Analysis II

Last updated Oct 24, 2020

Analysis, Volume II
Terence Tao
Hindustan Book Agency, January 2006. Third edition, 2014
Hardcover, 236 pages.

ISBN 81-85931-62-3 (first edition)

This is basically an expanded and cleaned up version of my lecture notes for Math 131B. In the US, it is available through the American Mathematical Society. It is part of a two-volume series; here is my page for Volume I.

— Errata to the first edition (softcover) —

p. 392, example 12.1.7: $5+2=7$ should be $3+4=7$ .
p. 393, example 12.1.9: $\sup(5,2)=7$ should be $\sup(3,4)=4$ .
p. 394, example 12.1.13: (iii) should be (c).
p. 403, example 12.2.13: delete the redundant “, but not the other”.
p. 404, line 4: “neither open and closed” should be “neither open nor closed”.
p. 415, line 3: $\alpha \in A$ should be $\alpha \in I$ .
p. 416, line 11: “ $k \geq j$ ” should be “ $k > j$ “.
p. 419, line -2: In Exercise 12.5.15, = should be $\neq$ . Also, “that by counterexample” should be “by counterexample that”
p. 426, Exercise 13.2.9: $X$ should be ${\Bbb R}$ throughout. Also, the definition of limsup and liminf for functions has not been given; it can be reviewed here, e.g. by inserting “where we define $\limsup_{x \to x_0} f(x) := \inf_{r>0} \sup_{x: |x-x_0| \leq r} f(x)$ and $\liminf_{x \to x_0} f(x) := \sup_{r>0} \inf_{x: |x-x_0| \leq r} f(x)$ .”
p. 435, Definition 13.5.6: “metric space” should be “topological space”.
p. 438, Exercise 13.5.9: One needs to assume as an additional hypothesis that X is first countable, which means that for every x in X there exists a countable sequence V_n of neighborhoods of x, such that every neighbourhood of x contains one of the V_n.
p. 452, Exercise 14.3.6: “Propositoin” should be “Proposition”.
p. 452, Exercise 14.3.8: “ $x \in {\Bbb R}$ ” should be “ $x \in X$ “.
p. 458: Exercise 14.5.2 should be deleted and redirected to Exercise 14.2.2(c).
p. 459: In line 11, $2 \epsilon$ should be $\epsilon$ .
p. 464: ) missing at the end of Exercise 14.7.2. An additional exercise, Exercise 14.7.3 is missing; it should state “Prove Corollary 14.7.3.”.
p. 466: Exercise 14.8.8 should be Exercise 14.8.2.
p. 467: Exercise 14.8.11 should be Exercise 14.8.4.
p. 469: “Limits of integration” should be “Limits of summation”. In Lemma 14.8.14, $3M+2\delta$ should be $1+4M$ , and Exercise 14.8.14 should be Exercise 14.8.6.
p. 470: Exercise 14.8.15 should be Exercise 14.8.7. Exercise 14.8.16 should refer to a (currently non-existent) Exercise 14.8.9, which of course would be to prove Lemma 14.8.16.
p. 471: At the end of the proof of Corollary 14.8.19, $|P(y)-g(y)| \leq \varepsilon$ should be $|P(y)-f(y)| \leq \varepsilon$ .
p. 472: In Exercise 14.8.2(c), Lemma 14.8.2 should be Lemma 14.8.8.
p. 477: In Exercise 15.1.1(e), Corollary 14.8.18 should be Corollary 14.6.2.
p. 478: In Example 15.2.2, $\frac{1}{x-1}$ should be $\frac{1}{1-x}$ .
p. 482: In Exercise 15.2.5, the $(x-b)^n$ on the right-hand side should be $(x-b)^m$ .
p. 486: In second and third display, y should be in ${}[0,1]$ rather than $(-1,1)$ .
p. 493: In Exercise 15.5.4, $x < 0$ should be $x \leq 0$ .
p. 501: In Theorem 17.7.2, “if $f(x_0)$ is not invertible” should be “if $f'(x_0)$ is not invertible”.
p. 502: In Exercise 15.6.6, Lemma 15.6.6 should be Lemma 15.6.11.
p. 511: “Fourier… was, among other things, the governor of Egypt during the reign of Napoleon. After the Napoleonic wars, he returned to mathematics.” should be “Fourier… was, among other things, an administrator accompanying Napoleon on his invasion of Egypt, and then a Prefect in France during Napoleon’s reign.”
p. 556: In Theorem 17.5.4, f can take values in ${\Bbb R}^m$ and not just in $R$ ; insert the line “By working with one component of $f$ at a time, we may assume $m=1$ ” as the first line of the proof. Also, $f(x-x_0)$ should be $f(x+x_0)$ .
p. 557: In the second display, $-f(0)$ should be $+f(0)$ .
p. 560: In Exercise 17.6.1, add the hypothesis “and $f'$ is continuous” before “, show that $f$ is a strict contraction”.
p. 561: In Exercise 17.6.3, change “which is a strict contraction” to “such that $|f(x)-f(y)| < |x-y|$ for all distinct $x,y$ in ${}[a,b]$ “. In Exercise 17.6.8, $\max(c,c')$ should be $\min(c,c')$ .
p. 562: In Theorem 17.7.2, $T: E \to {\Bbb R}^n$ should be $f: E \to {\Bbb R}^n$ .
p. 565, line -7: $U$ should be $f^{-1}(B(0,r/2))$ rather than $f^{-1}(B(0,r))$ .
p. 570, first display: all partial derivatives should have a – sign (not just the first one). Last paragraph: “Thus $(x_1,\ldots,x_{n-1})$ lies in W” should be “Thus $(x_1,\ldots,x_{n-1})$ lies in U”.
p. 571, second display: add “ $=0$ ” at the end.
p. 584, Corollary 18.2.7: “ $x_i \in [a_i,b_i]$ ” should be “ $x_i \in (a_i,b_i)$ “.
p. 599, Definition 18.5.9: $(a,\infty)$ should be $(a,+\infty]$ .
p. 600: In Lemma 18.5.10, $f: \Omega \to {\Bbb R}$ should be $f: \Omega \to {\Bbb R}^*$ . In the second and fourth lines of the proof of this lemma, $(a,+\infty)$ should be $(a,+\infty]$ .
p. 616-617, Exercise 19.2.10: ${\Bbb R}$ should be ${}[0,1]$ throughout.

— Errata to the second edition (hardcover) —

p. 351, At the end of Example 12.1.6, add “Extending the convention from Example 12.1.4, if we refer to ${\bf R}^n$ as a metric space, we assume that the metric is given by the Euclidean metric unless otherwise specified.”
p. 372, In Case 1 of the proof of Theorem 12.5.8, all occurrences of “ $y^{(n)}$ should be $y^{(n_j)}$ in the second paragraph.
p. 374, In Exercise 12.5.12(b), the phrase “with the Euclidean metric” should be deleted.
p. 390: In Exercise 13.5.5, “there exist $a, b \in X$ such that the “interval” $\{ y \in X: a < y < b \}$ ” should be replaced with “there exists a set $I$ which is an interval $\{y \in X: a < y < b\}$ for some $a, b \in X$ , a ray $\{y \in X: a < y \}$ for some $a \in X$ , the ray $\{ y \in X: y < b\}$ for some $b \in X$ , or the whole space $X$ , which”. In Exercises 13.5.6 and 13.5.7, “Hausdorff” should be “not Hausdorff”.
p. 390: Exercise 13.5.8 should be replaced as follows: “Show that there exists an uncountable well-ordered set $\omega_1+1$ that has a maximal element $\infty$ , and such that the initial segments $\{ x \in \omega_1+1: x < y \}$ are countable for all $y \in \omega_1+1 \backslash \{\infty\}$ . (Hint: Well-order the real numbers using Exercise 8.5.19, take the union of all the countable initial segments, and then adjoin a maximal element $\infty$ .) If we give $\omega_1+1$ the order topology (Exercise 13.5.5), show that $\omega_1+1$ is compact; however, show that not every sequence has a convergent subsequence.”
p. 395: In Proposition 14.1.5(d), add “Furthermore, if $x_0 \in E$ , then $f(x_0)=L$ .”
p. 396: In Exercise 14.1.5, $\lim_{y \to y_0; y \in f(E)} g(x)$ should be $\lim_{y \to y_0; y \in f(E)} g(y)$ , and $\lim_{x \to x_0; x \in E} g \circ f(x_0)$ should be $\lim_{x \to x_0; x \in E} g \circ f(x)$ .
p. 425: In Theorem 15.1.6(d), the summation should start from n=1 rather than n=0.
p. 427: Just before Definition 15.2.4, “for some $a \in {\ bf R}$ ” should be “for some $r > 0$ “.
p. 431: In Exercise 15.2.8(e), “ $d_m (x-b)^n$ ” should be “ $d_m (x-b)^m$ . In Exercise 15.2.8(d), $(s-\varepsilon)^m$ should be $(s-\varepsilon)^{-m}$ .
p. 433 (proof of Theorem 15.3.1): $s_N$ in the third display and $s_0$ in the next line should be
$S_N$ and $S_0$ respectively.
p. 452: In Exercise 15.7.2, $x-y$ should be $x_0-y$ . In Exercise 15.7.6, “complex real number” should be “complex number”.
p. 473: In Exercise 16.5.4, $in \hat f(n)$ should be $2\pi i n \hat f(n)$ .
p. 477: In Example 17.1.7, $x \in {\Bbb R}$ should be $c \in {\Bbb R}$ .
p. 486: In Definition 17.3.7, $1 \leq j \leq m$ should be $1 \leq j \leq n$ , and $x_0+tv$ should be $x_0+te_j$ .
p. 488: In the definition of L in the proof of Theorem 17.3.8, m should be n.
p. 492: In Exercise 17.3.1, Exercise 17.1.3 should be Exercise 17.2.1.
p. 495: In the proof of Theorem 17.5.4, $a'$ should equal $\frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j} f(0)$ rather than $\frac{\partial}{\partial x_j} \frac{\partial}{\partial x_i} f(0)$ .
p. 499, proof of Lemma 17.6.6: After “ $F$ does indeed map $B(0,r)$ to itself.”, add “The same argument shows that for a sufficiently small $\varepsilon > 0$ , $F$ maps the closed ball $\overline{B(0,r-\varepsilon)}$ to itself. After “ $F$ is a strict contraction”, add “on $B(0,r)$ , and hence on the complete space $\overline{B(0,r-\varepsilon)}$ “.
p.502, proof of Theorem 17.7.2: “ $f^{-1}(y)={\tilde f}^{-1}(y+f(x_0))$ ” should be “ $f^{-1}(y)={\tilde f}^{-1}(y-f(x_0))$ “.
p. 505, Section 17.8: $(0,1)$ should be $(-1,0)$ . In the second paragraph, the function $f$ should be $g$ (for better compatibility with the discussion of the implicit function theorem).
p.508, proof of Theorem 17.8.1, “U is open and contains $(y_1, \dots,y_{n-1},0)$ ” should be “U is open and contains $(y_1, \dots,y_{n-1})$ “.
p. 515: In the display before Definition 18.2.4, $j=\in J$ should be $j \in J$ . In Definition 18.2.4, $\sum_{j=1}^\infty$ should be $\sum_{j \in J}$ .
p. 520: In Example 18.2.9, $\sum_{q\in\mathbf{Q}} m*(\mathbf{Q})$ should be $\sum_{q\in\mathbf{Q}} m*(\{q\})$ in the display.
p. 528, proof of Lemma 18.4.8: On the second line, “let $A$ be any other measurable set” should be “let $A$ be an arbitrary set (not necessarily measurable)”.
p. 545: In Corollary 19.2.11, “non-negative functions” should be “non-negative measurable functions”.
p. 555, Remark 19.5.2: x and y should be swapped in “equals 1 when $x>0$ and y=0, equals -1 when $x<0$ and y=0, and equals zero otherwise”.

— Errata for the third edition (hardcover) —

Page 18: The proof of Lemma 1.4.3 should refer to Exercise 1.4.1, not Exercise 1.4.3.
Page 21: In Exercise 1.4.8(c), add “(This part of the exercise requires the axiom of choice.)”.
Page 22: In Definition 1.5.3, add “We call $(X,d)$ bounded if $X$ is bounded”.
Page 32: In Corollary 2.2.3(b), “ $f/g: X \rightarrow \mathbb{R}$ is also continuous at $x_{0}$ ” should be “ $f/g: X \rightarrow \mathbb{R}$ is also continuous”, and similarly for “ $cf: X \rightarrow \mathbb{R}$ are also continuous at $x_0$ “.
Page 42: In Exercise 2.5.5, $V \subset X$ should be $V \subseteq X$ . In Exercise 2.5.13, “topological space” should be “Hausdorff topological space”. In Exercise 2.5.4, after “the trivial topology is not Hausdorff” add “if the space contains at least two elements”.
Page 43: In Exercise 2.5.6, insert “and” between “if $x \in X$ ” and “ $(V_n)_{n=1}^\infty$ is any…”.
Page 48: In Exercise 3.1.5, $\lim_{x \in x_0; x \in E}$ should be $\lim_{x \to x_0; x \in E}$ .
Page 71: In Corollaries 3.8.18 and 3.8.19, “supported on $[0,1]$ ” is redundant and may be deleted.
Page 98, 100: Exercises 4.6.10-4.6.13 may be deleted, and the paragraph after Lemma 4.6.13 may be replaced with “Observe that with our choice of definitions, the space ${\bf C}$ of complex numbers is identical (as a metric space) to the Euclidean plane ${\bf R}^2$ , since the complex distance between two complex numbers $(a,b), (a',b')$ is exactly the same as the Euclidean distance $\sqrt{(a-a')^2+(b-b')^2}$ between these points. Thus, every metric property that ${\bf R}^2$ satisfies is also obeyed by ${\bf C}$ ; for instance, ${\bf C}$ is complete and connected, but not compact.”
Page 122: In Remark 5.5.2, “continuously differentiable” may be relaxed to just “differentiable”, and “twice continuously differentiable” may be relaxed to “continuously differentiable”.
Page 123: In the sixth line, “Corollary 5” should be “Corollary 5.3.6”.
Page 140: In the last paragraph, “continuous on $F$ ” should be “continuous at $x_0$ “.
Page 146: The definition of continuous differentiability needs to be detached from Definition 6.5.1 and placed near the beginning of this page.
Page 159: In the first line of the third display in the proof of Theorem 6.8.1, the $f$ ‘s should be $F$ ‘s.
Page 161: “if some other derivative $\frac{\partial f}{\partial x_j}$ is zero” should be “if some other derivative $\frac{\partial f}{\partial x_j}$ is non-zero”.
Page 176: In Proposition 7.3.3, $\sum_{q \in J} q+E$ should be $\bigcup_{q \in J} q+E$ (two occurrences).
Page 178: In Corollary 7.4.7, $m(B \backslash A) = m(B) - m(A)$ should be $m(B \backslash A) + m(A) = m(B)$ .

— Errata for the corrected third edition —

Page 10: In Exercise 1.1.8, a right parenthesis is missing at the end of the last sentence. In Exercise 1.1.11, ${\mathbf R}^n$ should be $X$ .
Page 16: In the first paragraph, the first parenthetical comment should be closed after “… and hence outside of $E$ .” In the second parenthetical comment, the period should be outside the parenthesis. “The point 0” should be “The point $(0,0)$ ” (two occurrences).
Page 22: In Definition 1.5.3, insert “for every $x \in X$ ” before “there exists a ball” (in order to keep the empty metric space bounded).
Page 23: In Theorem 1.5.8, $I$ should be $A$ in the statement of the theorem (four occurrences). In Case 2 of the proof, $B(y,r_0/2) \in V_\alpha$ should be $B(y,r_0/2) \subset V_\alpha$ . One should in fact split into three cases, $r_0=0$ , $0 < r_0 < \infty$ , and $r_0=\infty$ . For the last case, write “For this case we argue as in Case 2, but replacing the role of $r_0/2$ by (say) $1$ “. In the proof of Theorem 1.5.8, $Y \subset \bigcup_{\alpha \in F} V_\alpha$ should be $Y \subseteq \bigcup_{\alpha \in F} V_\alpha$ .
Page 26: In Exercise 1.5.10, $n$ should be a natural number rather than a positive integer (in order to ensure that the empty set is totally bounded).
Page 29: In Theorem 2.1.4(c), all occurrences of $\subset$ should be $\subseteq$ .
Page 30: In Exercise 2.1.7, $E \subset Y$ should be $E \subseteq Y$ .
Page 33: Add an additional Exercise 2.2.12 after Exercise 2.2.11: “Let $f: {\bf R}^2 \to {\bf R}$ be the function defined by $f(x,y) := x^2/y$ when $y \neq 0$ , and $f(x,y) := 0$ when $y = 0$ . Show that $\lim_{t \to 0} f(tx, ty) = f(0,0)$ for every $(x,y) \in {\bf R}^2$ , but that $f$ is not continuous at the origin. Thus being continuous on every line through the origin is not enough to guarantee continuity at the origin!”
Page 34: In Proposition 2.3.2, replace “Furthermore, ” with “Furthermore, if $X$ is non-empty”,
Page 37: In Theorem 2.4.5, replace “Let $X$ be a subset…” with “Let $X$ be a non-empty subset…”.
Page 43: Exercise 2.5.8 is incorrect (the space $\omega_1+1$ is sequentially compact) and should be deleted.
Page 44: In Exercise 2.5.14, add “Hausdorff” before “topological space”.
Page 46: In Definition 3.1.1, the domain of $f$ should be $E$ rather than $X$ . Similarly for Proposition 3.1.5, Exercise 3.1.3, and Exercise 3.1.5. In Remark 3.1.2, $\lim_{x \in x_0; x \in E} f(x)$ should be $\lim_{x \to x_0; x \in E} f(x)$ .
Page 47: In Proposition 3.1.5(c), all occurrences of $\subset$ should be $\subseteq$ .
Page 48: In Exercise 3.1.1, add the hypothesis “Assume that $x_0$ is an adherent point of $E \backslash \{x_0\}$ (or equivalently, that $x_0$ is not an isolated point of $E$ )”. In Exercise 3.1.3, replace the last three sentences with “If $X$ is a topological space and $Y$ is a Hausdorff topological space (see Exercise 2.5.4), prove the equivalence of Proposition 3.1.5(c) and 3.1.5(d) in this setting, as well as an analogue of Remark 3.1.6. What happens to these statements of $Y$ is not Hausdorff?”.
Page 56: In item (c) of Section 3.4, a right parenthesis is missing after Definition 3.2.1. In Definition 3.4.2, add the hypothesis that $X$ is non-empty, and add “uniform metric” next to “sup norm metric” and $L^\infty$ metric”. In Remark 3.4.1, “(b) is a special case of (a)” should be “(a) is a special case of (b)”.
Page 59: Before Definition 4.5.5, “exp is increasing” should be “exp is strictly increasing”.
Page 60: In Example 3.5.8, “ratio test” should be “root test”, and Theorem 7.5.1 should be “from Analysis I”.
Page 61: In the second to last display, the factor $2$ in front of $2 \varepsilon(b-a)$ should be deleted.
Page 62: In Example 3.6.3, Lemma 7.3.3 should be “from Analysis I”.
Page 64: At the end of the first paragraph, the period should be inside the parentheses.
Page 77: In Remark 4.1.9, it is more appropriate to add “uniformly” after “assures us that the power series will converge”.
Page 78: At the end of the Exercise 4.1.1, a right parenthesis should be added.
Page 82: In Exercise 4.2.3, the period should be inside the parentheses. In the first paragraph, a right parenthesis should be added.
Page 83: At the end of Exercise 4.2.8(e), the period should be inside the parentheses. Also in the hint, Fubini’s theorem should be Theorem 8.2.2 of Analysis I, and a remark needs to be made that one may also need to study an analogue of the $d_m$ in which the $c_n$ are replaced by $|c_n|$ .
Page 86: In the last two displays, $\limsup_{n \to \infty}$ should be $\limsup_{y \to 1; y \in [0,1)}$ .
Page 92: At the end of Exercise 4.5.1, a right parenthesis should be added.
Page 99: before the final paragraph, add “Inspired by Proposition 4.5.4, we shall use $\exp(z)$ and $e^z$ interchangeably. It is also possible to define $a^z$ for complex $z$ and real $a>0$ , but we will not need to do so in this text.”
Page 102: In the second paragraph parenthetical, the period should be inside the parentheses.
Page 103: In the second paragraph, a period should be added before “In particular, we have…”.
Page 105: In the last paragraph of Exercise 4.7.9, the period should be inside the parentheses (two occurrences).
Page 112: In Example 5.2.6, $\sin(x)$ should be $\sin(2\pi x)$ .
Page 113: In Exercise 5.2.3, “so that” should be “show that”. For more natural logical flow, the placing of Exercises 5.2.2 and 5.2.4 should be swapped.
Page 116: In Theorem 5.4.1, “trignometric” should be “trigonometric”. In the paragraph after Remark 5.3.8, the period should be inside the parenthesis.
Page 125: In the last sentence of Exercise 5.5.3, the period should be inside the parenthesis. In Exercise 5.5.4, add “Here the derivative of a complex-valued function is defined in exactly the same fashion as for real-valued functions.”
Page 129: In Example 6.1.8, “clockwise” should be “counter-clockwise”.
Page 133: At the end of the proof of Lemma 6.1.13, $1 \leq j \leq m$ should be $1 \leq j \leq n$ . Expand the sentence “The composition of two linear transformations is again a linear transformation (Exercise 6.1.2).” to “The composition $T \circ S$ of two linear transformations $T,S$ is again a linear transformation (Exercise 6.1.2). It is customary in linear algebra to abbreviate such compositions $T \circ S$ of linear transformations by droppinng the $\circ$ symbol, thus $T \circ S = TS$ .”
Page 135: In the first paragraph, the period should be inside the parenthesis.
Page 138: In Example 6.3.3, “the left derivative” should be “the negative of the left derivative”. In the last sentence, the period should be inside the parenthesis.
Page 139: In the second paragraph, second sentence, the period should be inside the parenthesis; also in the final sentence. Expand the third display to “ $\frac{\partial f}{\partial x_j}(x_0) = D_{e_j} f(x_0) = -D_{-e_j} f(x_0) = f'(x_0) e_j$ , and expand “From Lemma 6.3.5” to “From Lemma 6.3.5 (and Proposition 9.5.3 from Analysis I)”.
Page 140: In the beginning of the proof of Theorem 6.3.8, $L(v_j)_{1 \leq j \leq m}$ should be $L(v_j)_{1 \leq j \leq n}$ , and similarly the sum on the RHS should be from $1$ to $m$ rather than from $1$ to $n$ .
Page 141: The period in the last line (before “and so forth”) should be deleted.
Page 142: At the end of the page, $(\sum_{j=1}^n v_{j}\frac{\partial f_i}{\partial x_j}(x_0))_{i=1}^m$ should be $(\sum_{j=1}^n v_{j}\frac{\partial f_i}{\partial x_j}(x_0))_{1\leq i\leq m}$ .
Page 144: In Exercise 6.3.2, $D_{e_j} f(x_0) = D_{-e_j} f(x_0)$ should be $D_{e_j} f(x_0) = -D_{-e_j} f(x_0)$ .
Page 146: In the second paragraph, third sentence, the period should be inside the parenthesis.
Page 148: In the proof of Clairaut’s theorem, $|x| \leq 2\delta$ should be $\| x\| \leq 2\delta$ .
Page 151: In Exercise 6.6.1, the range of $f$ should be $[a,b]$ rather than ${\bf R}$ .
Page 153: In the second paragraph of the proof of Theorem 6.7.2, “ $f(x_0)$ is not invertible” should be “ $f'(x_0)$ is not invertible”.
Page 154: In the last text line, $f(x)-x$ can be $g(x)=f(x)-x$ for clarity.
Page 155: In the proof of Theorem 6.7.2, after the display after “we have by the fundamental theorem of calculus. add “where the integral of a vector-valued function is defined by integrating each component separately.”
Page 156: $V - 0$ should be $V - \{0\}$ . The definition of $U$ should be $f^{-1}(V) \cap B(0,r)$ rather than $f^{-1}(B(0,r/2))$ (and the later reference to $U = f^{-1}(V)$ can be replaced just by $U$ ).
Page 157: In the final paragraph of Section 6.7, “differentiable at $x_0$ ” should be “differentiable at $f(x_0)$ “. Add the following Exercise 6.7.4 after Exercise 6.7.3: “Let the notation and hypotheses be as in Theorem 6.7.2. Show that, after shrinking the open sets $U, V$ if necessary (while still having $x_0 \in U$ , $f(x_0) \in V$ of course), the derivative map $f'(x)$ is invertible for all $x \in U$ , and that the inverse map $f^{-1}$ is differentiable at every point of $V$ with $(f^{-1})'(f(x)) = (f'(x))^{-1}$ for all $x \in U$ . Finally, show that $f^{-1}$ is continuously differentiable on $V$ .”
Page 158: In the first paragraph, final sentence, the period should be inside the parentheses.
Page 161: Add the following Exercise 6.8.1: “Let the notation and hypotheses be as in Theorem 6.8.1. Show that, after shrinking the open sets $U,V$ if necessary , that the function $g$ becomes continuously differentiable on all of $U$ , and the equation (6.1) holds at all points of $U$ .”
Page 164: In the first paragraph of Section 7.1, $\Omega \subset {\bf R}^n$ should be $\Omega \subseteq {\bf R}^n$ .
Page 165: Superfluous period in Theorem 7.1.1.
Page 169: After (7.1), $\prod_{j=1}^n [a_i,b_i]$ should be $\prod_{i=1}^n [a_i,b_i]$ .
Page 170: In the first paragraph “For all other values if $x$ ” should be “For all other values of $x$ “.
Page 172: $\subset$ should be $\subseteq$ (three occurrences).
Page 173: In Exercise 7.2.2, final sentence: period should be inside parentheses. Also, add “Here we adopt the convention that $c \times +\infty = +\infty \times c$ is infinite for any $0 < c \leq +\infty$ and vanishes for $c = 0$ .” In Example 7.2.12, “countable additivity” should be “countable sub-additivity”.
Page 174: In the penultimate paragraph, “identical or distinct” should be “identical or disjoint”, and $\subset$ should be $\subseteq$ . Also, “coset of ${\bf R}$ ” should be “coset of ${\bf Q}$ “; in the next paragraph, “the rationals ${\bf R}$ ” should be “the rationals ${\bf Q}$ “.
Page 175: In the second paragraph, “constrution” should be “construction”. After the third paragraph, add “Note also that the translates $q+E$ for $q \in {\bf Q}$ are all disjoint. For, if there were two distinct $q,q' \in {\bf Q}$ with $q+E$ intersecting $q'+E$ , then there would be $A,A' \in {\bf R}/{\bf Q}$ such that $q+x_A = q'+x_{A'}$ . But then $A = x_A + {\bf Q} = x_{A'} + {\bf Q} = A'$ and thus $x_A = x_{A'}$ , which implies $q=q'$ , contradicting the hypothesis.”
Page 178: In Lemma 7.4.5, “and any set $A$ ” should be “then for any set $A$ “.
Page 180: In the first paragraph, “Lemma 7.4.5” should be “Lemma 7.4.4(d)”. Also, in the display preceding this paragraph, enclose the sum in parentheses in the middle and right-hand sides (so that the supremum is taken over the sum rather than just the first term).
Page 187: In Example 8.1.2, the period should be inside the parentheses in the first parenthetical, and the final right parenthsis should be deleted.
Page 188: In Lemma 8.1.5 the function $f$ should take values in $[0,+\infty]$ rather than ${\bf R}$ (and then the requirement that $f$ be non-negative can be deleted).
Page 189: In the parenthetical sentence before Remark 8.1.8, the period should be inside the parentheses. In the first display in Lemma 8.1.9, the right-hand side summation should be up to $N$ rather than $n$ , and “are a finite number” should be “be a finite number”
Page 190: In the final display in the proof of Lemma 8.1.9, an equals sign should be inserted to the left of the final line.
Page 194: In Theorem 8.2.9, $f_n$ should take values in $[0,+\infty]$ rather than ${\bf R}$ .
Page 196: Before the second display, Proposition 8.2.6(cdf) should be Proposition 8.2.6(bce). Also add “It is not difficult to check that the $E_n$ are measurable”. In the first paragraph, all instances of $\subset$ should be $\subseteq$ .
Page 197: After the third display. Proposition 8.1.9(b) should be Proposition 8.1.10(bd).
Page 200: In the hint to Exercise 8.2.10, the “for all $n \geq N$ ” should be moved inside the set builder notation $\{ x \in [0,1]: f_n(x) > 1/m \}$ , thus using $\{ x \in [0,1]: f_n(x) > 1/m \hbox{ for all } n \geq N \}$ instead.
Page 201: Before Definition 8.3.2, when Corollary 7.5.6 is invoked, add “(which can be extended to functions taking values in ${\mathbf R}^*$ without difficulty)”.
Page 202: In the start of the proof of Theorem 8.3.4, add “If $F$ was infinite on a set of positive measure then $F$ would not be absolutely integrable; thus the set where $F$ is infinite has zero measure. We may delete this set from $\Omega$ (this does not affect any of the integrals) and thus assume without loss of generality that $F(x)$ is finite for every $x\in \Omega$ , which implies the same assertion for the $f_n(x)$ .
Page 204: In the second display, $\leq A - \frac{1}{n}$ should be $\geq A + \frac{1}{n}$ instead.
Page 205: In Proposition 8.4.1, add the hypothesis that $I$ is bounded.
Page 206: In the last paragraph, last sentence, the period should be inside the parentheses. In the last two displays, $\Omega$ should be $I$ .
Page 207: In the third paragraph, “Secondly, we could fix” should be “Thirdly, we could fix”.
Page 208: In the last paragraph, Lemma 8.1.4 should be Lemma 8.1.5.

Caution: the page numbering is not consistent across editions. In the third edition, the chapters were renumbered to start from 1, rather than from 12.

Thanks to Quentin Batista, Biswaranjan Behera, José Antonio Lara Benítez, Dingjun Bian, Petrus Bianchi, Philip Blagoveschensky, Carlos, cebismellim, EO, Florian, Aditya Ghosh, Gökhan Güçlü, Yaver Gulusoy, Minyoung Jeong, Bart Kleijngeld, Eric Koelink, Wang Kunyang, Brett Lane, Matthis Lehmkühler, Zijun Liu, Rami Luisto, Jason M., Manoranjan Majji, Geoff Mess, Jorge Peña-Vélez, Cristina Pereyra, Issa Rice, SkysubO, Rafał Szlendak, Winston Tsai, Kent Van Vels, Andrew Verras, Murtaza Wani, Xueping, Sam Xu, Zelin, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections.

337 comments

Comments feed for this article

23 November, 2007 at 10:37 am

Anonymous

Dear Terry,

The AMS url in your post links to another book rather than “Analysis II.” Additionally, I was trying to order the book from Amazon, and was only able to order “Analysis I” through Amazon directly, but couldn’t do the same for “Analysis II”, even though a page exists for the second volume. Any plans on it being sold *directly* by Amazon?

Thanks!

23 November, 2007 at 12:02 pm

Terence Tao

Dear Anonymous,

Thanks for the bad link report. I just checked Amazon and the second volume is currently available (well, there are two copies available, at least):

http://www.amazon.com/gp/product/8185931631/ref=pd_cp_b_0?pf_rd_p=317711001&pf_rd_s=center-41&pf_rd_t=201&pf_rd_i=8185931623&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=0ES3B9XEHE3S5E7CKHCS

But it is a rather low volume item, so availability is likely to continue to be spotty.

20 February, 2009 at 4:30 am

carlos

Dear Professor Tao,

First of all I would like you to congratulate you for the two volumes of the book. I had been looking for a reference to revise calculus from scratch and they have proven invaluable.

I would like to point out a few other minor errata I noticed:

p. 471 at the end of the proof |P(y)-g(y)| should be |P(y)-f(y)|
p.477 ex. 15.1.1 corollary 14.6.2 seems a better hint for (e)
p.482 ex.15.2.5 at the end of the identity, the last exponent should be m and not n

I also have a small question concerning Ex. 15.1.1 in page 477. To prove (c) the hint is to use the Weierstrass M-test. However, in the case of a function with an infinite convergence radius (such as for example sin(x) ), the sup norm of each of the f_n is infinite and consequently their infinite sum is not convergent, so I don’t see how the test can be applied. I am probably just making some basic mistake and I would really appreciate it if someone could point it out…

20 February, 2009 at 9:09 am

Terence Tao

Dear Carlos,

Thanks for the corrections! For Ex. 15.1.1(c), one is only asking for uniform convergence on a compact interval [a-r,a+r] (r can be arbitrarily large, but is finite), and in this case the summands are absolultely convergent.

22 March, 2009 at 10:37 am

Florian

Dear Professor Tao,

firstly, I can only agree with Carlos and I congratulate you to your book as well. I especially like the structure of it because you emphasize the importance of the basic elements of analysis (which aren’t obvious at all…).

I have a question concerning your example on page 625. Is the given example function f(x,y) really not (absolutley) integrable over y in x=0?
It would make sense to me, if it wasn’t integrable over x in y=0. Or am I wrong?

Best regards,
Florian

24 March, 2009 at 2:05 pm

Terence Tao

Oops, you’re right, it should be switched. Thanks!

22 April, 2009 at 10:17 pm

Mark

Professor Tau,
I am attempting exercise 12.5.14 in Analysis II and would like to know if you could elaborate on the hint given.

Any help would be greatly appreciated.

Thank you.
Mark

7 March, 2013 at 10:00 pm

Luqing Ye

Dear Mark,

I provide my hint in 2013,though your question is in 2009…my hint is essentially same to Mr.Tao’s hint,but I express it differently.

Because for all the point $x$ in E , $d(x_0,x)$ is non-negative,so the set of all $d(x_0,x)$ has a greatest lower bound $l$ which is non-negative.If this greatest lower bound can be reached by a point $p$ in E,such that $d(p,x_0)=l$ ,then done.Otherwise,【there is a sequence of elements $m_1,m_2,\cdots,m_n,\cdots$ in E,such that $\lim_{n\to\infty}d(m_n,x_0)=l$ 】.Because E is compact,so this sequence has a convergent subsequent,suppose that this subsequence convergent to $k$ in E,now we prove that $d(k,x_0)=l$ .If not,then $d(k,x_0)>l$ ,then this contradict to the words in 【】.

23 April, 2009 at 12:16 pm

Terence Tao

Dear Mark,

By compactness, the sequence $x^{(n)}$ in the hint has a subsequence which converges to a limit $x$ in E. Now compute what the distance of x to $x_0$ is.

7 March, 2013 at 10:25 pm

Luqing Ye

Dear Prof.Tao,

The only difference between my hint and your hint is that you introduce $d(x_0,x^(n))\leq R+\frac{1}{n}$ ,but I make some skip,I directly say $\lim_{n\to\infty}d(m_0,x_0)=l$ .I think this skip make my argument requires the axiom of choice….

No……I have to rethink,I think that your argument also require the axiom of choice…we need the axiom of choice to construct that sequence.

Is that true?

8 March, 2013 at 9:54 pm

Luqing Ye

Maybe I need to answer my question myself.I hope my comments do not disturb you.

Actually,lemma 8.4.5 has already answered this question.When E is a closed set,we don’t need the axiom of choice.Prof.Tao’s argument do not require the axiom of choice,my argument is nothing,because I really make some skip!I need to explain why the words in 【】is true.

23 April, 2009 at 5:51 pm

Mark

Dear Professor Tau,

Thank you for taking the time to assist me.
Your help has been greatly appreciated.

Mark

30 April, 2009 at 9:15 am

Jorge

Dear Professor Tao,

I’m trying to crack Ex. 14.8.2. in your Analysis II but I’m stuck on the part c), namely on the estimation|c*(1 – x^2)^N| <= epsilon which must hold for delta <= |x| <= 1 (this is precisely the condition from part c) in the definition of the approximation to the identity, I mean…). How can I now use part b) of the same Ex. here? Or this is not the case?

Thank you.
Jorge

30 April, 2009 at 10:06 am

Terence Tao

Dear Jorge,

Part (b) ensures that the normalising constant c only grows polynomially in N (or more precisely, is bounded by $C \sqrt{N}$ for some C). On the other hand, once x is bounded away from zero, $(1-x^2)^N$ decays exponentially in N (uniformly in x). As exponential decay always beats polynomial growth, the integral of $c (1-x^2)^N$ away from zero will go to zero.

30 April, 2009 at 10:36 pm

Jorge

So that f_N(x) := c(1-x^2)^N goes to zero function uniformly as N tends to infinity?

1 May, 2009 at 8:54 am

Terence Tao

On any set of the form $\{x: \varepsilon \leq |x| \leq 1 \}$ , yes.

1 May, 2009 at 9:15 am

Jorge

Thank you very much for your time. It is really great thing to be able to consult with you online.

Jorge

23 May, 2009 at 6:31 am

HASSAN JOLANY

HELLO I AM STUDENT UNIVERSITY OF TEHRAN I STUDY ABOUT THE POINCARE CONJECTURE FOR MY THESIS CAN I ASK YOU SOME OF MY PROBLEMS
BEST REGARD FOR YOU

1 June, 2009 at 10:09 pm

Joe

Dear Professer Tao,
I am attempting exercise 14.2.1 (a) from Analysis II. What do I need to show? For the “only if” direction I think I need to show that if f is continuous and the sequence goes to zero then I have pointwise convergence (is this right?), but I am not sure about the “if” direction. Can you give me any help? Thank you,

– Joe

2 June, 2009 at 8:42 am

Mark

Dear Prof Tao,

Is there any way I could get or browse the answers for all the exercises in Analysis 2?

2 June, 2009 at 9:43 am

Terence Tao

Dear Joe,

Yes, this is the “only if” part. The “if part” asserts that if f is a function such that $f_{a_n}$ converges pointwise to f for every sequence $a_n$ converging to zero, then f is necessarily continuous.

Dear Mark: I do not have an answer key for the exercises. Some of the exercises are based on my classes at

http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/

which have a partial solution set, but this is far from complete (also, some of the wording of the exercises changed when converted to book form).

2 June, 2009 at 5:05 pm

mark

Thank you prof. Another thing is there any solutions for all the homeworks of your Math131BH?

2 June, 2009 at 7:28 pm

Ali Nesin

Hi prof,

I do come across this q and really need your advise.
Suppose (X,d1), (X,d2) are two metric spaces over the same set X, and c1, c2 are strictly positive.
How do we show that (X,d), where d = c1d1 + c2d2 is also metric space?

2 June, 2009 at 8:44 pm

timur

Dear Ali Nesin,

Try going over the metric space axioms d has to satisfy. Which axiom(s) are you having trouble with?

3 June, 2009 at 1:02 am

Ali Nesin

I do have trouble actually in understanding the q. I think it is the real line, R I’m dealing with. Am I right?

3 June, 2009 at 1:18 pm

timur

Dear Ali Nesin,

I think you should ask your question on a forum like the following (I hope prof. Tao’s does not mind):

http://www.sciforums.com/

http://www.mathlinks.ro/Forum/

You even might find your question already answered.

5 June, 2009 at 6:31 pm

Mark

Mark

Professor Tau,
I am attempting exercise 14.8.8 in Analysis II and would like to know if you could elaborate on the hint given.

Any help would be greatly appreciated.

Thank you.
Mark

6 June, 2009 at 2:12 pm

Anonymous

In the proof of Abel’s Theorem on page 487 is the limit superior suppose to be as y approaches 1 instead of as n approaches infinity?

6 June, 2009 at 8:12 pm

Mark Werner

This is Mark Werner.

I am the original ‘Mark’ in this blog who asked about question 12.5.14. I am not the most recent ‘Mark’ asking about question 14.8.8 which has been set as a take home exam question with no consultation allowed. I believe someone in the class is pretending to be me, since the question was asked in exactly the same way as my original one, including the misspelling your name. I am posting this just to ensure that my name is clear.

Thank you.

– Mark

7 June, 2009 at 10:29 am

student

Dear Prof Tao,

is it possible to find a 1-1 map from [0,1] to (0,1)?

thanks

7 June, 2009 at 10:44 am

timur

Dear student,

There is a general theorem in set theory (called Cantor-Bernstein theorem) that says that for two sets A and B if there are injective maps A -> B and B -> A then there exists a bijective map A -> B. I believe Cantor proved it using AC and later Bernstein gave a constructive proof without AC.

7 June, 2009 at 2:39 pm

student

Thank you Timur,

actually I want to apply that theorem, there is an obvious 1-1 map from (0,1) to [0,1] but I could not find a 1-1 map from [0,1] to (0,1).

if anyone knows, please let me know

thanks

7 June, 2009 at 3:07 pm

timur

Sorry, I misunderstood your question; you can scale and translate [0,1] to, say, [z,1-z] for some z>0 to get an injective map [0,1] -> (0,1).

7 June, 2009 at 3:15 pm

student

good idea thanks Dear Timur

7 June, 2009 at 7:41 pm

geometry beginner

can anyone suggest me a nice introductory book on differential geometry and algebraic topology? my background is just point set topology.

if you know nice introductory survey articles as well, please let me know.

thanks

18 July, 2009 at 5:02 pm

Jason M.

Hello — I had a couple of very small observations:

On page 546, in Definition 17.3.7 (Partial derivative), $f$ is a function defined on a subset of $R^n$, and you are defining the partial derivative with respect to $x_j$, but you index $j$ from 1 to $m$ instead of from 1 to $n$.

Later in that same definition, you write $\lim_{t\to 0;t\neq 0;x_0+tv\in E}$, when I think it ought to be $x_0+te_j$ instead of $x_0+tv$.

Thanks! –Jason

18 July, 2009 at 5:49 pm

Jason M.

Hello again — I have noticed one more thing:

On page 548, in the proof of Theorem 17.3.8, when you are defining the linear transformation L, you index j from 1 to m instead of 1 to n.

These are pretty small details, but I think they could cause confusion for someone unfamiliar with the material!

Thanks! –Jason

[Thanks for the corrections – T.]

20 March, 2010 at 10:31 am

Matt

Dear Professor Tao:

On page 520 of Analysis II, it states that the Fourier Transform is a map from functions in C(R/Z;R). Should the codomain of the functions be the complex numbers? It seems inconsistent with the rest of the surrounding definitions and theorems inasmuch as it only applies to functions with a real codomain. Thank you very much.

Matt

24 June, 2010 at 1:31 pm

Guilherme

Dear Prof. Tao,

I really liked your book. The contents of both volumes are very very good.
I was thrilled. I was thinking omg, this will be my favourite analysis book.

But I saw that you place the proof of some theorems and lemmas as exercises. I really have huge difficulties to learn from books which use this style.

When this happens, it seems that suddenly the author of the book “abandoned me”. And then I get lost. Of course, it a good habit of doing the proofs by myself, but, in the end, I am not sure my proof is good/rigorous or not.

But, the organization and the writing style are amazing !

Best,
Guilherme

24 June, 2010 at 1:41 pm

Terence Tao

This is very much an intentional choice on my part (see page xiii of the preface). There are a lot of important subtleties in real analysis (e.g. the distinction between pointwise convergence and uniform convergence, to give just one example), and the only way to really appreciate this is to chew through some of the foundational results regarding these subtleties. There are some parts of mathematics where it is sufficient to just know the main results and their proofs, as opposed to deriving them with one’s own mental resources, but in my view analysis is not one of these subjects.

24 June, 2010 at 2:09 pm

Guilherme

Dear Prof. Tao,

It is very nice to have this feedback about learning methods from you. And it is indeed true that I learn better when I put some effort in trying to do the proof on my own before seeing the proof from the actual book.

When this happens, I appreciate much more the proof and understand better the overall(big picture) strategy.

On the other hand, what happens if the student doesnt know how to make the proof, or if the student doesnt know if the his proof is wrong ? Don’t you think that the student will miss an important part of the material ?

ps: I am not making this questions with the intention to criticize. Mainly, I am making this questions in order to improve my learning methods too.
I always get frustrated when I realized that the author puts the proofs in the exercises. Maybe it is time for a change. :)

Best,
Guilherme

24 June, 2010 at 2:30 pm

Terence Tao

A student that is unable to make proofs, or to verify whether a given proof is correct, will have difficulty with a subject as theory-intensive as real analysis regardless of whether the textbook contains proofs of its theorems or not, particularly when one reaches the point where one needs a result not already contained in the textbook.

In the case of my own textbook, in most cases when a proof of a theorem is set as an exercise, a proof of a related or similar result is provided in the text as a model; the ability to adapt an argument from one context to another is another important skill in analysis that is difficult to pick up by any means other than direct practice.

10 October, 2012 at 4:54 am

Luqing Ye

I am in the contrary.I learn mathematics by proving theorems.Though in this way I learn math slowly,i quite enjoy it ,that’s enough.In learning Prof.Tao’s book(and other people’s book),I just prove everything I encounter,it is very often that I spend several days proving a difficult theorem,that doesn’t matter,time is used to waste.When a theorem is proved by me,and I believe that my proof is correct,I will write down my proof,and put it in my personal blog(If a theorem is proved by me,I will not read author’s proof,I have no patient absorbing other people’s idea).Why I write down my proof ?Because I like to record things,if it is not recorded,several weeks later,it may go like wind,on the other hand,if a theorem is proved by me ,maybe the proof is original,original things must be recorded.Record my own proof also helps me to review,maybe in the future I will forget what I learned(that’s very often for me),then I will review my proof of the theorems in that field,then I will quickly recall nearly everything,by the way,in the process of reviewing,I will read my proof seriously the second times to check whether it has some mistake in it .That’s how I learn mathematics,maybe in the future I should adjust my method a bit,it is very tired to do every myself.

31 March, 2013 at 9:29 am

Misha Shvartsman

Following that logic, it would not make sense to have any homework exercise that requires a proof. And, indeed, these days browsing internet you probably can find any proof you want, but chance that you learn anything will not be great :)

24 June, 2010 at 6:24 pm

Tom

Dear Terry,

Can one use matrices and linear algebra to explain both pointwise and uniform convergence?

9 August, 2010 at 5:17 am

Bart Kleijngeld

It’s only a minor correction, but they all count, right? On page 536, example 17.1.7 you write $x \in \mathbb{R}$ , while it should be $c \in \mathbb{R}$ .

Thanks for your great book,
Bart

[Corrected, thanks – T.]

7 September, 2010 at 7:30 pm

Student

Dear Prof. Tao, I am having trouble with 13.2.10 in analysis 2. I have to show that
dx((x,y), (x0, y0)) is greater than dx((x,y0), (x0, y0)) (holding one variable constant). It seems rather obvious, but I cannot seem to get it.

7 September, 2010 at 7:36 pm

Terence Tao

That’s not quite the right approach. I recommend that you write down in full (using epsilons and deltas) what it means for the two-variable function $(x,y) \mapsto f(x,y)$ to be continuous at $(x_0,y_0)$ , and what it means for the function $x \mapsto f(x,y_0)$ to be continuous at $x_0$ , taking particular care to get all the quantifiers (“for all”, “there exists”) correct and in the right order. Once you have both definitions down side by side, it should become clear how to proceed.

14 September, 2010 at 3:11 pm

Juan

Hello Dr. Tao,

I am having trouble with 13.4.6 in your book, Analysis II. I have been trying to do a proof by contradiction, assuming that the union of an arbitrary number of connected sets is in fact disconnected, but I am stuck.

Here is what I have:

Call the arbitrary union of the connected sets, E.
Assume that there exist open, disjoint, nonempty sets A, B, such that A union B = E.

Then A = complement of B.

Ea (the alpha’th set in the collection of connected sets). Assume Ea is not contained in A nor B (rather it is “split up” among the two sets). Thus there exists a subset of A, call it D, and a subset of B, call it E, s.t. D union E = Ea. But Ea is connected, so D and E are either nondisjoint, or one of the sets is not open. That is where I am stuck. What do I do from there? I assume that I want to show that for all alpha, Ea must be contained in either A or B, so then we get a contradiction, but I do not know.

14 September, 2010 at 7:56 pm

Terence Tao

If A is open in E, then $D := A \cap E_\alpha$ will be open in $E_\alpha$ .

It may help to isolate a common point $x_0$ of the intersection $\bigcap_\alpha E_\alpha$ , which is non-empty by hypothesis. Note that $x_0$ must belong to exactly one of A or B.

4 November, 2010 at 9:04 am

Kent Van Vels

Hello Dr. Tao,

I am going through your analysis book. I may have found a typo in the second edition. In exercise 16.5.4, should the formula be

$\widehat{f'(n)} = i n \widehat{f}(n)$ be changed to

$\widehat{f'(n)} = 2\pi i n \widehat{f}$ ?

Thanks,

Kent

[Added, thanks – T.]

4 November, 2010 at 9:06 am

Kent Van Vels

I mean, of course,

$\widehat{f'}(n) = 2\pi i n \widehat{f}(n)$ .

13 November, 2010 at 11:10 am

Kent Van Vels

I have a question the second edition of Analysis II.
For the proof of Proposition 14.1.5 (exercise 14.1.2)

Can you explain where my reasoning goes wrong in showing that the third statement is logically distinct from the first statement?

If we consider the function $f:\mathbb{R}\to \mathbb{R}$ defined as $f(x) = 0$ for $x\not =0$ , and $f(x) = 1$ for $x=0$ . Then
$\lim_{x\to 0; x\in \mathbb{R}} f(x) = 0$ .
Now, let $V=(-1/2,1/2)$ , an open set which contains $0$ . Now, every open set that contains $0$ will contain an interval of the form $(-a,b)$ where $a,b>0$ . But $f((-a,b) \cap \mathbb{R}) = \{0,1\} \not \subset (-1/2,1/2)= V$

Do we need to assume that $f$ is continuous or that $x_{0}$ is a limit point (instead of an adherent point)?

Thanks,

Kent Van Vels

13 November, 2010 at 12:14 pm

Terence Tao

For your specific function, the limit $\lim_{x \to 0; x \in {\bf R}} f(x)$ does not exist, because you are not excluding 0 from the range of possible values of x. (This notational convention may be a bit different from what is seen in some other texts; see Remark 14.1.2.)

13 November, 2010 at 1:11 pm

Kent Van Vels

Thank you. That clears up my confusion.

23 November, 2010 at 2:55 pm

Anonymous

Prof. Tao,

After studying the Chapter 17 Several variable differential calculus in Analysis vol.2 , I am curious about whether there are high order derivatives for the function $f: R^n\rightarrow R^m$ . But a further consideration seems to tell me that the high order derivative is not the transformation as $L_{Df}$ any more. It seems that one has to define the “second derivative” for the transformation $L_{Df}:R^n\rightarrow R^{m\times n}$ . But it seems strange.

Could you explain about this? If they do exist, what’s the topic there in the larger picture? Or is this concept meaningless?

23 November, 2010 at 8:12 pm

Terence Tao

These higher order derivatives are best described either as multilinear maps or as tensors. For instance, the second derivative is a bilinear map that is essentially the Hessian. The third and higher derivatives can also be defined, but they are less pleasant to work with, basically because multilinear algebra is significantly less pleasant than linear algebra in many ways. (Also, when working on a curved domain, the underlying curvature of the space begins to assert its influence much more strenuously on the third and higher derivatives than on the first and second ones, leading to further technical complications.)

30 November, 2010 at 11:09 am

Kent Van Vels

In Lemma 18.4.8 (the proof of countable additivity of Lebesgue measure), The third sentence that begins “Let $A$ be any other measurable set…”. Is it ok to only consider measurable sets $A$ to show that $E$ measurable?

[Oops, $A$ should be an arbitrary set rather than a measurable one. I’ve added an appropriate erratum – T.]

1 December, 2010 at 2:50 pm

Kent Van Vels

Can you explain where my reasoning goes wrong in showing a
counterexample to Exercise 13.5.5?

Let $X$ be the set $\{0,1\}\subset \mathbb{N}$ , with the $0\prec 1$ . Then $X$ is totally ordered. But for
$x=0$ (an element of $X$ ) we can’t find an $a\in X$
such that $a\prec x$ . This shows
that $X$ isn’t open, and so $(X,\mathcal{F})$ isn’t a
topology.

1 December, 2010 at 4:16 pm

Terence Tao

Oops, one also needs to allow for half-rays and the whole space in addition to intervals. I’ve added an appropriate erratum.

1 December, 2010 at 7:32 pm

Kent Van Vels

Should Exercise 13.5.6 and Exercise 13.5.7 be changed to read that the topologies in question are non-Hausdorff? It seems impossible for two disjoint open sets to exist.

[Corrected, thanks – T.]

2 December, 2010 at 11:00 am

KVV

In Proposition 12.4.12 it says

“Let $(X,d)$ be a metric space…” should this be changed to read

“Let $(X,d)$ be a complete metric space?

Thanks.

2 December, 2010 at 11:17 am

Terence Tao

X needs to be complete for part (b), but this is not needed for part (a).

4 December, 2010 at 11:01 pm

Kent Van Vels

Thanks. That clears up my misunderstanding.

Kent

4 December, 2010 at 12:59 pm

KVV

In exercise 12.5.12 there is a hint that reads in part,

“…useless here since that only applies to Euclidean spaces with the Euclidean metric.” Should that be changed to indicate that the Heine-Borel theorem also applies to Euclidean spaces equipped with the sup-norm and taxi-cab metric?

Thanks,

Kent

[Corrected, thanks – T.]

12 December, 2010 at 10:52 am

KVV

On page 515 in the second edition there might be two small typos.

There is math display that reads, $latex\cdots \sum_{j=\in J}\cdots$, I don’t think the equals sign is required.

Also, in the definition of the outer measure, there is a sum from $j=1$ to $\infty$ . Should this be a sum over $j\in J$ ?

Thanks,

Kent

[Corrected, thanks – T.]

13 December, 2010 at 8:24 am

KVV

Hello,

On page 545, in the statement of Corollary 19.2.11, there is a sequence of functions. Do we have to assume that each function in the sequence is measurable?

Thanks,

Kent

[Corrected, thanks – T.]

29 December, 2010 at 1:07 pm

KVV

I think there is a small typo in the statement of Exercise 15.2.8. For part (e), there is a sum whose summand is $d_m (x-b)^n$ , I think it should probably be $d_m (x-b)^m$ .

Thanks.

[Corrected, thanks – T.]

17 January, 2011 at 3:22 pm

KVV

I think there are two small typos in the statement of exercise 14.1.6 on page 396, (second edition).

First, there is a display $\lim_{y\to y_0; y\in f(E)} g(x) = z_0$ , I think it should probably be $\lim_{y\to y_0;y\in f(E)} g(y) = z_0$ .

Second, there is a display $\lim_{x\in x_0; x\in E} g \circ f(x_0) = z_0$ , I think it should probably be $\lim_{x\to x_0; x\in E} g\circ f(x) = z_0$

Thanks

[Corrected, thanks – T.]

13 March, 2011 at 10:56 pm

Biswa

Dear Prof. Tao,

On Page 478, just before Definition 15.2.4, “…and differentiable on (a-r, a+r) for some $a\in{\mathbb R}$” should be “…and differentiable on (a-r, a+r) for some $r>0$.

Thank you.
Biswaranjan Behera

[Corrected, thanks – T.]

15 March, 2011 at 3:26 am

Biswa

P 485, $s_N$ (third display) and $s_0$ (next line) should be
$S_N$ and $S_0$ . Also, should it be “by convention
$S_0=-D$ in place of “in particular $S_0=-D$ ?

[Corrected, thanks. The way summation has been defined in the text, an empty summation is automatically zero without the need for any explicit further convention. -T.]

30 April, 2011 at 1:52 pm

Anonymous

I am confused by the exercise 14.1.1. Why “the limit $lim_{x\to x_0;x\in E}f(x)$ exists iff the limit $lim_{x\to x_0;x\in E\setminus \{x_0\}}f(x)$ AND is equal to $f(x_0)$ ?” This is true only when $f:E\to Y$ is continuous, isn’t it?

30 April, 2011 at 11:55 pm

Terence Tao

No, continuity is not required for this fact. (Both the hypothesis and the conclusion of the exercise, though, are equivalent to $f$ being continuous at $x_0$ .)

1 May, 2011 at 8:43 pm

Anonymous

Dear Prof. Tao,

I am not sure if this is the right place to put the following question here.

In the very beginning of the book, Chapter 1 of volume I, you introduce what is real analysis and point out that it is the theoretical foundation “which underlies calculus”. However, I didn’t find that you talk about the “multivariable calculus” in you real analysis series courses, from math131AB to math 245ABC.

Actually, you did talk about the “several variable differential calculus” in Chapter 17 in this volume. But why there is no “multivariable integration” here? Is it because it is enough for the theory which you have talked about in your series books(Analysis I,II; An introduction to measure theory;An epsilon of room, Vol I) to build up the “multivariable integration”, or because it is beyond the scope of this course?

Since I’d like to build up the system of knowledge about real analysis myself, I follow the series of your books. But it puzzles me that I am even not able to find the “multivariable integration” in your books.

1 May, 2011 at 9:34 pm

Terence Tao

See Chapter 19.

1 May, 2011 at 9:55 pm

Anonymous

I did not put my question in the right way. I think I am talking about the [multiple integral](http://en.wikipedia.org/wiki/Multiple_integral) and the [vector analysis](http://en.wikipedia.org/wiki/Vector_calculus). In the one dimensional case, you talked about the Riemann integral in Chapter 11 and finally the Lebesgue integration in Chapter 19. In the n-dimensional case, you introduce the Lebesgue integration in ${\bf R}^n$ in Chapter 19. So there is no “Riemann integral” but the Lebesgue integration in the multi-dimensional case? What’s the theoretical counterpart of multiple integral in the vector analysis in your book?

2 May, 2011 at 7:08 am

Terence Tao

See Section 19.5.

Generalising the one-dimensional Riemann integral as set up in Chapter 11 to higher dimensions would be a good exercise for a student who has gone through that chapter. (I explicitly set that exercise, incidentally, as Exercise 1.1.26 of my measure theory book.) However, I feel that this is something that a good student can work out on his or her own, after understanding the core topics covered in texts such as my own (as well as seeing a lower-division treatment to higher-dimensional integration).

4 June, 2011 at 3:36 pm

Anonymous

I find that you didn’t deal with the “Local maxima, local minima” in the multivariate case in these two volumes. Is this intentional? Will you recommend any references which deal with this topic rigorously?

12 June, 2011 at 8:08 am

Anonymous

Errata to the second edition, ‘formula does not parse’

[Corrected, thanks – T.]

12 June, 2011 at 9:23 am

Anonymous

Dear Prof. Tao,
I am confused with the definition of partial derivative(p.546 Definition 17.3.7). Is it a special case of the “directional derivatives”? If it is, why the constrain for $t$ in the definition is $t\to 0;t\neq 0$ instead of $t\to 0;t>0$ which appears in the definition of directional derivative? If it is not, how can you conclude from Lemma 17.3.5 that $\frac{\partial f}{\partial x_j}(x_0)=f'(x_0)e_j$ ?

12 June, 2011 at 10:49 am

Terence Tao

The partial derivative $\frac{\partial f}{\partial x_j}$ is related to the two directional derivatives $D_{e_j} f$ and $D_{-e_j} f$ in that whenever $D_{e_j} f(x)$ and $-D_{-e_j} f(x)$ exist and are equal to each other, then the partial derivative $\frac{\partial f}{\partial x_j}(x)$ also exists and is equal to these two quantities, and conversely (this follows from Proposition 9.5.3).

12 June, 2011 at 7:15 pm

Anonymous

Dear Prof. Tao,

For the Exercise 17.3.3, by definition, $D_vf(0.0)=\frac{v_1^3}{v_1^2+v_2^2}$ where $v=(v_1,v_2)$ . and ${\partial}{\partial x}f(0,0)=1$ , ${\partial}{\partial y}f(0,0)=0$ . Then by your remark in p.547, $D_vf(0,0)=\sum_jv_j\frac{\partial f}{\partial x_j}(0,0)$ , $f$ is not differentiable at $(0,0)$ . Is the proof supposed to be like this? Or do you have any hint for this?

[This will work, yes – T.]

12 June, 2011 at 7:19 pm

Anonymous

Dear Prof. Tao,
I didn’t see any connection between Exercise 17.3.1 and Exercise 17.1.3. (You said that “this will be similar to Exercise 17.1.3”)Typo?

[Sorry, this should be Exercise 17.2.1. -T.]

15 June, 2011 at 9:20 am

Chris

Dear Professor Tao

I had another confusion that require clarification.

1) In your above errata, you mention, Pg 478: Just before definition, ………. should be ………..

My comment: The actual page is 427 but I can’t find the one that need to be replace. Can you please justify this?

2) As to your statement on Pg 485, ………….

My Comment: The correction can be made by me because the page number is wrong and I can’t identify where is that part in the text that need to be replaced.

3) Pg 548 you mention that m should be n. Can you clarify further as to what should be replaced in the text.

4) Pg 593 & Pg 625 .<— For this errata, there is no such pages in your second edition book. Can you justify which should be corrected in the second edition as I can't locate what need to be corrected.

[All errata for the first edition should already be corrected for the second edition – T.]

16 June, 2011 at 7:12 pm

Anonymous

In the section 17.8: The implicit function theorem, p.568, the notation is a little confusing for me. In the first paragraph, $f:{\bf R}^n\to{\bf R}$ gives rise to a graph $\{(x,f(x)):x\in{\bf R}^n\}$ in ${\bf R}^{n+1}$ . And in the second paragraph, $f$ is used in “a hypersurface of the form $\{x\in{\bf R}^n:f(x)=0\}$ , where $f:{\bf R}^n\to {\bf R}$ is some function”. Are these two $f$ the same?? I guess the $f$ in the theorem 17.8.1 is the same as that in the second paragraph, while $g$ is the one in the first paragraph? Am I right?

[Yes, that is right; I will change the first f to a g to reduce confusion. -T.]

9 August, 2011 at 2:56 am

Anonymous

p. 563, line 10: “We remark that …if $f(x_0$ is” should be “We remark that …if $f'(x_0$ is”.

Biswa

[Added, thanks – T.]

9 August, 2011 at 9:48 pm

Anonymous

p. 560, Lemma 17.6.6, Can one apply the contraction mapping theorem to F as it is defined on B(0, r)?

[Oops; I’ve adjusted the argument there accordingly. -T.]

9 August, 2011 at 9:59 pm

Anonymous

p. 567, line -2, (0, 1) should be (-1, 0).

[Added, thanks – T.]

12 August, 2011 at 9:21 pm

Anonymous

p.563, line 17, ‘ $f^{-1}(y)={\tilde f}^{-1}(y+f(x_0))$ ‘ should be ‘ $f^{-1}(y)={\tilde f}^{-1}(y-f(x_0))$ ‘..

p.570, second line of the last paragraph, ‘U is open and contains $(y_1, \dots,y_{n-1},0)$ ‘ should be ‘U is open and contains $(y_1, \dots,y_{n-1})$ ‘.

Biswa

8 February, 2012 at 9:02 pm

Anonymous

Can anyone show me if this integral $\int_{\pi}^{3\pi/2}\frac{1}{x^2 \sin^{3/2}(x)}dx$ is finite or not?

Thanks

5 May, 2012 at 5:50 am

ugroh

Terence, two comments on Analysis 2:

p.25 Theorem 15.1.6. (d): Should not the formula for $f'$ starts with
$n=1$ instead of $n=0$ ? This would be more consistent with Prop. 15.2.6. and the formula there.

p.427, 3rd line from the top: … differentiable on .. for some $r \in R$ instead of $a \in R$ ?

Ulrich

[Corrections added, although I will have to recheck the page numbers on the errata list the next time I have access to a physical copy of the book, as some of the numbers seem to be incorrect.]

6 May, 2012 at 9:58 pm

ugroh

Sorry, it should be p. 425 not p. 25
Ulrich

25 July, 2012 at 5:24 am

Gökhan Güçlü

Dear Professor,

In the proof of Theorem 12.5.8 Case 1, in the second paragraph (As before, we know that there exists…) all the sequences should be subsequences

[Correction added, thanks -T.]

10 October, 2012 at 6:35 am

Luqing Ye

Dear Prof.Tao,

At the beginning of section 17.7,you say:

“We recall the inverse function theorem in single variable calculus(Theorem 10.4.2), which asserts that if a function $f:\mathbf{R} \to \mathbf{R}$ is invertible, differentiable, and $f'(x_0)$ is non-zero, then $f^{-1}(x)$ isdifferentiable at $f(x_0)$ , and $(f^{-1})'(f(x_0))=\frac{1}{f'(x_0)}$ .”

I think maybe you lost a condition……I think the condition “ $f^{-1}$ is continuous at $f(x_0)$ ” should be added.Theorem 10.4.2 has that condition.

10 October, 2012 at 10:37 am

Luqing Ye

Oops,It is my fault.At the beginning of section 17.7,it is $f:\mathbb{R}\to \mathbb{R}$ ,in this case, $f^{-1}:\mathbb{R}\to \mathbb{R}$ have to be continuous automatically.But in theorem 10.4.2,it is $f:X\to Y$ , $X,Y$ are arbitrary point sets. …..Oh my god,awesome mathematics……Sorry for my fault.

20 October, 2012 at 4:59 am

Matthis Lehmkühler

Dear Prof. Tao,

in Example 18.2.9. (p. 520 Second Edition, Hardcover), there is a little mistake in the middle part of the inequation: You have to replace $\sum_{q\in\mathbf{Q}} m*(\mathbf{Q})$ by $\sum_{q\in\mathbf{Q}} m*(\{q\})$ as mentioned in the above text.

25 October, 2012 at 11:16 am

Matthis Lehmkühler

Dear Prof. Tao,

in the proof of Theorem 17.5.4 (p. 495f. Second Edition, Hardcover) it sais that one has to proof that $a=a'$ whereby $a$ is defined to be $a:=\frac{\partial}{\partial x_j}\frac{\partial f}{\partial x_i}(0)$ and $a'$ should denote the quantity $a':=\frac{\partial}{\partial x_j}\frac{\partial f}{\partial x_i}(0)$ . Of course one has to replace $a':=\frac{\partial}{\partial x_j}\frac{\partial f}{\partial x_i}(0)$ by $a':=\frac{\partial}{\partial x_i}\frac{\partial f}{\partial x_j}(0)$ .

[Added, thanks – T.]

1 November, 2012 at 9:11 am

Anonymous

Dear Prof. Tao,
In exercise 14.1.2(prove Proposition 14.1.5), I think statement (d) isn’t logical
equivalent with (a) when $x_0\in E$ and $f(x_0)\neq L$ . such as:
$x_0=0$ , $E=[-1,1]$ , $f(x)=0(x\neq 0)$ , $f(0)=1$ , $L=0$ , then $g(x)=0$ is continuous at $0$ , but the limit $\lim_{x\rightarrow 0;x\in[-1,1]}f(x)$ does no exist.

[Yes, you’re right; I’ve added an erratum accordingly. -T.]

9 January, 2013 at 5:37 am

Jack

The power series can be generalized to complex functions. Can the Weierstrass approximation also true in complex analysis, say every complex continuous function in a closed disc in ${\bf C}$ can be uniformly approximated by polynomials? (I don’t see an obvious way to translate the proof in terms of complex functions.)

9 January, 2013 at 9:31 am

Terence Tao

If one considers polynomials of both $z$ and $\overline{z}$ , then one can recover the Weierstrass approximation theorem in the complex case; but if one restricts attention to polynomials of $z$ only, then the Weierstrass approximation theorem fails (one can approximate functions that are analytic in a neighbourhood of one’s domain, but not general continuous functions). See Section 3 of https://terrytao.wordpress.com/2009/03/02/245b-notes-12-continuous-functions-on-locally-compact-hausdorff-spaces/ for more discussion.

20 January, 2013 at 4:35 pm

Jack

Can one get from Theorem 15.1.6 (c) that $\sum_{n=0}^{\infty}c_n(x-a)^n$ converge uniformly to $f$ on the open interval $(a-R,a+R)$ ?

I saw several theorems about uniform convergence on a compact set or a closed bounded interval. And in complex analysis, there are also similar results such as “if $f$ is a holomorphic function in a disc $D$ , then it has a power series expansion that converges uniformly on every compact set in $D$ “.

Why is the uniform convergence on compact sets so special?

21 January, 2013 at 8:22 am

Jack

After some thought, I managed to come up with $\sum_{n=0}x^n$ as a counterexample. And the uniform convergence on compact sets is discussed in detail in 245b Notes 11(https://terrytao.wordpress.com/2009/02/21/245b-notes-11-the-strong-and-weak-topologies/). [Since I’m note able to edit the original questions (as people usually do in MO), I have to put the further words in a new comment box. Merging them together might be good though.]

The 245ABC series notes seem to be very complete regarding fundamental real analysis that one should be able to find an answer in them regarding basic questions of real analysis, although the author didn’t mention what’s “not” in detail (or not covered) in these notes :-)

7 March, 2013 at 5:55 am

Luqing Ye

Dear Prof.Tao,

In exercise 13.3.5,you say:Let $(X,d_X)$ be a metric space,and let $f:X\to \mathbf{R}$ and $g:X\to\mathbf{R}$ be uniformly continous functions,show that the direct sum $f\bigoplus g:X\to\mathbf{R}^2$ defined by $f\bigoplus g(x):=(f(x),g(x))$ is uniformly continous.

I think it is necessary to equip $\mathbf{R}^2$ with the Euclidean metric and $\mathbf{R}$ with the standard metric.Or maybe you have already said it earlier but I didn’t notice…

7 March, 2013 at 6:34 am

Luqing Ye

Oh my god,in Example 12.1.4 you have said without special remark we equip $\mathbf{R}$ with standard metric.But I can’t find where you said without special remark we equip $\mathbf{R}^2$ with Euclidean metric…

[A remark to this effect has been added to the errata, thanks. -T.]

9 March, 2013 at 10:41 pm

Luqing Ye

Dear Professor Tao,

Exercise 13.5.1 asks me to prove that when $\#X>1$ ,then the trival topology can not be obtained by placing a metric on $X$ .But I think this conclusion is not correct.

For example,if $X=\{p,q\}$ ,and $p\neq q$ ,then if we place a discret metric $d$ on X,then $d(p,p)=d(q,q)=0,d(p,q)=d(q,p)=1$ ,let $\tau=\{\emptyset,X\}$ ,then obviously $(X,\tau)$ is a trival topolggy.

So I obtained a trival topology by placing a metric on X ！

It is almost true that my argument might be wrong,the key point is what does “obtained ” mean..

10 March, 2013 at 12:51 pm

Terence Tao

“obtained” is in the sense in the paragraph after Definition 13.5.1, namely one needs to take the topology to be the collection of sets that are open with respect to the metric. In your example, the open sets with respect to the discrete metric are $\{ \emptyset, \{p\}, \{q\}, X \}$ , which is not the trivial topology.

10 March, 2013 at 9:04 pm

Luqing Ye

Thanks a lot.By the way,in your errata you say “In Exercises 13.5.6 and 13.5.7, “Hausdorff” should be “not Hausdorff”.”

This makes me amazing!Because yesterday I proved they are Hausdorff!

So it make me think what is a proof……I can prove what I believe,even though that is wrong…And I can’t prove what I don’t believe…

Oh my god…Prove rigourously is easier said than done.

10 March, 2013 at 3:48 am

Luqing Ye

Dear Prof.Tao,

An errata of your errata.

p. 390: In Exercise 13.5.5, “there exist a, b \leq X such that the “interval” \{ y \in X: a < y < b \}” should be replaced with “there exists a set I which is an interval \{y \in X: a < y < b\} for some a, b \in X, a ray \{y \in X: a < y \} for some a \in A, the ray \{ y \in X: y < b\} for some a \in A, or the whole space X, which”.

It should be

p. 390: In Exercise 13.5.5, “there exist $a, b \leq X$ such that the “interval” $\{ y \in X: a < y < b \}$ ” should be replaced with “there exists a set I which is an interval $\{y \in X: a < y < b\}$ for some $a, b \in X$ , a ray $\{y \in X: a < y \}$ for some $a \in A$ , the ray $\{ y \in X: y < b\}$ for some $b \in A$ , or the whole space X, which”.

I hope the errata of the errata of your errata won't exist.

[Errata corrected, thanks – T.]

10 March, 2013 at 11:54 pm

Luqing Ye

Dear Professor Tao,and everyone,

Sorry I have so many comments(They are not posted casually),but I think I found a counterexample to Exercise 13.5.8.

Exercise 13.5.8 is stated as follows:

Let X be an uncountable set,and let $\infty$ be an element of X.Let $\tau$ be the collection of all subsets E in X which are either empty or are co-countable and countain $\infty$ .Show that $(X,\tau)$ is a compact topological space…

But I think $(X,\tau)$ is not neccessarily a compact topological space.

The counterexample is as follows,this counterexample make use of the ordinals.

Lets see a set of ordinal numbers which are uncountable,they are $1,2,3,4,\cdots,n,\cdots,\omega,\omega+1,\omega+2,\cdots$ (According to the well ordering principle,an uncountable sets $X$ can be well ordered and thus order isomorphic to an ordinal).

Let $\omega=\infty$ ,Now we see the subsets $P_0=\{\omega,\omega+1,\cdots,\omega+n,\cdots\}$ , $P_1=P\bigcup\{1\}$ ,$\cdots$, $P_n=P_{n-1}\bigcup \{n\}$ .

It can be easily verified that $P_0,P_1,\cdots,P_n,\cdots$ is an open covering of X,and $\forall i\in\mathbf{N}$ , $P_i$ contains the element $\infty$ ( $\omega$ ),And it is also easy to verify that a finite subsets of $\{P_0,P_1,\cdots,P_n,\cdots\}$ can not cover X.Thus X is not a compact topological space.

Maybe there are mistakes in my arguement,I will appreciate any comments.If there is no mistake in my argument,then I really find a counterexample!r

11 March, 2013 at 8:02 am

Terence Tao

Sorry, the topology given in the exercise (the neighbourhood topology of the cocountable topology at infinity) was not the one intended; it is the order topology on $\omega_1+1$ which gives the required counterexample (namely, a topology that is compact but not sequentially compact). The replacement exercise is now given in the errata.

14 March, 2013 at 3:05 am

Luqing Ye

Dear professor Tao,

I can’t prove this new exercise,because I think I find a counterexample.The difficulty is that I cannot express my counterexample very clearly by using English,but I will try that.

Firstly,I will define a term “gap”.To understand what does “gap” mean,Lets see these ordinals:

$0,1,2,\cdots,n,\cdots,\omega,\omega+1,\cdots$

We call there is a gap between a natural number and the ordinal $\omega$ .If you start from a natural number $n$ and begin to walk step by step in the positive direction,one second a step,then you will never go out of that gap.

Now I begin to introduce my counterexample .

$\omega_1+1$ is an ordinal,for an element $x$ in $\omega_1+1$ ,if $x$ is a successor ordinal,now we prove that the set $\{x\}$ is an open set in the context of the order topology ,because if $x=\infty(\omega_1+1)$ ,then $x\in \{p\in \omega_1+1:p>\omega_1\}$ ,and $\{p\in\omega_1+1:p>\omega_1\}\subseteq \{\infty\}$ ;If $x\neq\infty$ ,then $x\in\{p\in\omega_1+1:x-1<p<x+1\}$ ,and $\{p\in\omega_1+1:x-1<p<x+1\}\subseteq \{x\}$ .

So when $x$ is a successor ordinal,I just cover $x$ by the open set $\{x\}$ .

If $x$ is a limit ordinal,and there are countable infinite gaps before $x$ (Obviously,such $x$ exists),then we cover $x$ by the set $\{p\in\omega_1+1:a<p\leq x\}$ ,where $a$ is a successor ordinal,and there are countable infinite number of gaps before $a$ (Obviously,such $a$ exists).Now we prove that $\{p\in\omega_1+1:a<p\leq x\}$ is an open set that covers $x$ .For $y\in \{p\in\omega_1+1:a<p\leq x\}$ ,if $y=a$ ,then $y\in \{p\in\omega_1+1:a-1<y<a+1\}$ ,and $\{p\in\omega_1+1:a-1<y<a+1\}\subseteq \{a\}$ ;if $y=x$ , $y\in \{p\in\omega_1+1:a<y<x+1\}$ ,and $\{p\in\omega_1+1:a<y<x+1\}\subseteq \{p\in \omega_1+1:a<p\leq x\}$ .So $\{p\in\omega_1+1:a<p\leq x\}$ is an open set that covers $x$ .So when $x$ is a limit ordinal and there are countable infinite gaps before $x$ ,I just cover it by the open set $\{p\in\omega_1+1:a<p\leq x\}$ .

It is easy to verify that there are countable infinite number of limit ordinals which has the above feature(countable infinite gaps before it),so it is easy to verify that we have constructed an infinite open cover which does not have finite open cover.

Done.

Maybe my counterexample is wrong,or maybe I do not express it well(But I examine my comments several times before it is posted).I will appreciate any comments.

14 March, 2013 at 5:55 am

Luqing Ye

A supplement to my comment:

$\omega_1+1$ here is a special “ordinal”,because I regard the maximal element $\infty$ in $\omega_1+1$ as $\omega_1+1$ itself.This doesn’t matter.

14 March, 2013 at 6:59 am

Luqing Ye

No,No,I cannot do something which is contradict the axiom of regularity.My supplement is wrong.

Dear prof.Tao,I am in mess,I am a newbie to such thing as ordinals…Could you please give me some hints to your new exercise,because now I even doubt the correctness of your new exercise……I think it is not compact,I can construct a open cover which has no finite subcover.

15 March, 2013 at 12:22 am

Luqing Ye

My counterexample is wrong because of the lack of knowlege in ordinal type.After see a brief introduction to ordinals in wikipedia,now I understand where I am wrong…

20 September, 2014 at 2:29 am

Anonymous

Hi, Luqing Ye, can you please show me how to prove the new exercise? in fact i have no idea to that exercise.

20 March, 2013 at 11:42 pm

Luqing Ye

Dear Prof.Tao,

In definition 14.8.4,you say “A function R–> R is said to be compactly supported iff it is supported on some interval [a,b]”

What does “some” mean?Some means “Infinite” or “finite(more than nothing)”,Or “nothing”?

21 March, 2013 at 1:10 am

Luqing Ye

OH,maybe here “some ” means “a”,because you write “interval” instead of “intervals”.But if it means “a”,then the concept of compactly supported function is same to the concept of the supported function…

21 March, 2013 at 12:57 pm

Anonymous

I m basing my answer only on your question here, I do not have the book at hand.

But it should be understood like this:

A function R -> R is said to be compactly supported, if there exists an interval [a,b], such that supp f \subset [a,b].

21 March, 2013 at 8:01 pm

Luqing Ye

Thanks.So if a function is compactly supported,then this function is supportted,and if this function is compactly supported,then this function is also supported.

Then why we need the definition of compactly supported?It is equivalent to the definition of supported……

[The definition on the book is following:
Let [a,b] be an interval.A function f:R–>R is said to be supported on [a,b] iff f(x)=0 for all $x\not\in [a,b]$ .We say that f is compactly supported iff it is supported on some interval [a,b].

]

22 March, 2013 at 1:04 am

Anonymous

Hello, the first definition is not that of a function being supported, but of a function being supported on a concrete interval [a,b]!

The definition says: Let [a,b] be an interval.A function f:R–>R is said to be !! supported on [a,b] !! …

I hope that clearifies your question!

22 March, 2013 at 1:36 am

Shen Zeng

Just in case my answer was too short, here is an example:

Take for example the characteristic function of [0,1], i.e. f(x) = 1 iff x in [0,1] and 0 otherwise.

This function is supported on [0,1]. It is supported also on [0,2], but not on [2,3]. Lastly, it is compactly supported, because it is supported for at least one interval [a,b], i.e. a=0 and b=1 or a=0 and b=2. This is because it f is supported on [0,1] and f is supported on [0,2].

I hope the difference and connections between these definitions became clear. If not, feel free to ask again!

22 March, 2013 at 1:55 am

Luqing Ye

Thanks for your explanation and your example,now I understand all…

【supported on some interval [a,b]】means 【there exists a interval [a,b] such that f is supported on [a,b]】

Maybe I have to go back to primary school……

27 March, 2013 at 11:49 pm

Luqing Ye

Hi,Prof.Tao,

In exercise 15.2.8,(d),you say: $|d_m|\leq C(s-\varepsilon)^m$ .But in (b),you say $|c_n|\leq C(r-\varepsilon)^{-n}$ .

I can’t prove (d),so I wonder whether $C(s-\varepsilon)^m$ should be changed into $C(s-\varepsilon)^{-m}$ …in that way,(b) and (d) would have the same form and will looks more comfortable…..

[Errata added, thanks – T.]

28 March, 2013 at 9:45 pm

Luqing Ye

Here is my proof.I can’t proceed any further.

———————————————————–
$\displaystyle |d_m|=|\sum_{n=m}^{\infty}\frac{n!}{m!(n-m)!} (b-a)^{n-m} c_n|\leq\sum_{n=m}^{\infty}|\frac{n!}{m!(n-m)!}(b-a)^{n-m}||c_n|\leq\sum_{n=m}^{\infty}C|\frac{n!}{m!(n-m)!}(b-a)^{n-m}(r-\varepsilon)^{-n}|=C\sum_{n=m}^{\infty}\frac{n!}{m!(n-m)!}|b-a|^{n-m}(r-\varepsilon)^{-n}$

According to Exercise 15.2.7,

$\displaystyle \sum_{n=m}^{\infty}\frac{n!}{m!(n-m)!}|b-a|^{n-m}(r-\varepsilon)^{-n}=\frac{r-\varepsilon}{(r-\varepsilon-|b-a|)^{m+1}}$

$\displaystyle |d_m|\leq C \frac{r-\varepsilon}{(r-\varepsilon-|b-a|)^{m+1}}=C \frac{r-\varepsilon}{r-\varepsilon-|b-a|}(r-\varepsilon-|b-a|)^{-m}$

We know that $r-|b-a|\geq s$ ,so $r-|b-a|-\delta=s$ , $\delta\geq 0$ .So

$\displaystyle |d_m|\leq C \frac{r-\varepsilon}{r-\varepsilon-|b-a|}(s+\delta-\varepsilon)^{-m}$ .

——————————————————–

So I doubt that you should let $s$ be as large as possible,i.e,let $s=r-|a-b|$ ,so that $\delta=0$ ,then I proved it.If $s$ is not big enough,I can’t prove this result…..

In fact,I think it doesn’t matter that we let $s=r-|b-a|$ .

2 April, 2013 at 7:37 am

Terence Tao

$(s+\delta-\varepsilon)^{-m} \leq (s-\varepsilon)^{-m}$ .

2 April, 2013 at 7:38 am

Luqing Ye

Hi Mr.Tao,

In Exercise 15.7.2,I think “Show that there exists a c>0 such that f(y) is non-zero whenever 0<|x-y|<c” should be “Show that there exists a c>0 such that f(y) is non-zero whenever 0<|x_0-y|<c”.

Sorry that my comment is not always right,so it needs your wisdom to judge which is right and which is wrong….

[Errata added, thanks – T.]

8 April, 2013 at 10:02 pm

Luqing Ye

Dear Prof.Tao,

This is not an errata,but a personal advice.In lemma 16.4.6,you make use of Fejer kernel to construct an $(\varepsilon,\delta)$ approximation to the identity.Why Fejer kernel?I think a long time to find the geometric meaning of the Fejer kernel,but the geometric meaning is rather complex,I make use of a sophiscated but wrong model of planet motion found by ancient astronomers to give Fejer kernel a geometric explanation(A circle A_2 moves around a circle A_1,and a circle A_3 moves around the circle A_2,etc…).

So I think the Fejer kernel approach is rather unintuitive to a begginer in Fourier analysis.Here is an intuitive approach:

Consider the function $f_n(x)=\cos^n \frac{\pi}{2}x,x\in [0,1]$ .When $n$ become larger and larger, $f_n(x)$ will become thiner and thiner,and at last will become a $(\varepsilon,\delta)$ approximation to the identity(In order to do this ,we have to multiply a constant $c$ to f_n(x),so that $\int_{[0,1]}cf_n(x)dx=1$ ).And it is easy to verify by using algebra of trigonometrics like $2\cos^2x-1=\cos 2x$ ,we can change $f_n(x)$ into trigonometric polynomials,that is done!

9 April, 2013 at 12:17 am

Luqing Ye

Oh!This method fails!In this method,I only construct an $(\varepsilon,\delta)$ approximation to the identity,not an $(\varepsilon,\delta)$ periodic approximation to the identity!(But by this wrong method,I can prove that a series of trigonometric polynomials can approximate uniformly to a continuous function on an interval.)

But even though this method is wrong,I think I will be right only after some minor corrections.Instead of $f_n(x)=\cos^n \frac{\pi}{2}x$ ,I need to construct a trigonometric function,this function is always non-negative,and at the place of $x=1$ ,this function should become large again.

Then let this function replace $\cos \frac{\pi}{2}x$ ,then I think that will be OK.

12 April, 2013 at 3:24 am

ugroh

Terence, on Page 452 (Analysis II), Exercise 15.7.6 you are writing
“.. be a non-zero complex real number ..”. I guess the “real” can be ignored.
Regards
Ulrich

17 April, 2013 at 8:27 am

Luqing Ye

Dear Prof.Tao,

I think the definition of derivative in this book (and in Spivak’s “calculus on manifold”,etc.) is inconsistent.

When we say “A differentiable function $f:\mathbb{R}^m\to \mathbf{R}^n$ has derivative $L$ at point $x_0\in \mathbf{R}^m$ “,we regard $L$ as a linear map from $\mathbf{R}^m$ to $\mathbf{R}^n$.

But in the case of $m=n=1$ ,we regard $L$ as a real number.A real number is a real number,not a linear map,isn’t it?(Though a real number multiplying a stuff means a linear map )

It is very similar to such conditions,that is,the distinction between the matrix and the correspoinding linear map.Maybe somebody regard matrix as a linear map(I don’t know whether “somebody” exists or not,I just guess),but I regard matrix as a matrix, $m$ columns, $n$ rows.Only when a matrix multiplying a stuff forms a linear map…

17 April, 2013 at 9:07 am

Terence Tao

In one dimension, there is a canonical isomorphism that identifies each real number $L$ with the associated dilation map $x \mapsto Lx$ on the reals, so one customarily “abuses notation” by identifying the two (somewhat analogously to how one identifies natural numbers with a subset of the integers, or the rationals as a subset of the reals, etc.).

In higher dimensions, the corresponding identification between matrices and linear transformations is dependent on the choice of basis, which is why it is important to keep the two concepts separate. But in one dimension the identification is basis-independent, and there is little harm in conflating the two concepts (or, for that matter, with identifying $1 \times 1$ matrices with scalars).

17 April, 2013 at 9:34 am

Luqing Ye

Dear Prof Tao,

I just now say “Hmm,this comment is wrong again”.After seeing your reply,It seems that my comment is not wrong again,because in your book is

$\displaystyle \frac{f(x)-f(x_0)-L(x-x_0)}{|x-x_0|}$ ,In spivak’s book is $\displaystyle \frac{f(x)-f(x_0)-h(x)}{|x-x_0|}$ .

Now I understand,thanks for your reply!

17 April, 2013 at 9:11 am

Luqing Ye

Hmm…This comment is wrong again…In fact,when $m=n=1$ ,the linear transformation is $h(x):f'(x_0)(x-x_0)$ ,as illustrated in Spivak’s book…

23 May, 2013 at 10:16 pm

BWW

Dear Professor Tao,

Should the second inequality in (12.1) on page 392 read $d_{l_1}(x,y)\leq nd_{l^2}(x,y)$ instead of $d_{l_1}(x,y)\leq \sqrt{n}d_{l^2}(x,y)$ ? I think $d_{l^1}((.4,.4),(0,0))=.8\geq \sqrt{2}d_{l^2}((.4,.4),(0,0))$ is a counterexample to the original inequality.

24 May, 2013 at 2:08 am

Luqing Ye

I don’t think so.Cauchy’s inequality can’t be wrong.

In your counterexample, $d_{l^1}((.4,.4),(0,0))=.8= \sqrt{2}d_{l^2}((.4,.4),(0,0))$ ,which does not conflict $d_{l_1}(x,y)\leq \sqrt{n}d_{l^2}(x,y)$ .

13 July, 2013 at 4:15 am

Andre

Exercise 14.1.1. Is the requirement that the if you are assume that the limits are equal to $f(x_0)$, then the equivalence holds ? So you require that $f$ is continuous ?

13 July, 2013 at 7:12 am

Terence Tao

Not quite. The first part of the question asks to show, under the hypotheses stated, that “ $\lim_{x \to x_0; x \in E} f(x)$ exists” is true if and only if “ $\lim_{x \to x_0; x \in E \backslash \{x_0\}} f(x)$ exists AND this limit is equal to $f(x_0)$ ” is true. The second part of the question asks to show that “ $\lim_{x \to x_0; x \in E} f(x)$ exists” implies that “ $\lim_{x \to x_0; x \in E} f(x)$ is equal to $f(x_0)$ “. In neither case is continuity of $f$ assumed, although if any of the above premises are true then it is indeed the case that $f$ is necessarily continuous at $x_0$ when restricted to $E$ .

13 July, 2013 at 9:32 am

Andre

Thank’s a lot.

5 October, 2013 at 11:41 am

Anonymous

Do we really need $f$ being countinuously differentiable to recover the pointwise convergence of the Fourier series to $f$ as you stated in Remark 16.5.2? A local result states that if $f$ being an integrable function on the circle is differentiable at a point $\theta_0$ , then the Fourier series converges to $f$ at that point. Is it because we need global pointwise convergence that the countinuity of the derivative is needed?

5 October, 2013 at 12:50 pm

Terence Tao

Hmm, you’re right; continuous differentiability may be relaxed to just differentiability for the purposes of this remark.

5 October, 2013 at 4:49 pm

Jack

Can continuous differentiability of $f$ actually give absolute and uniform convergence of the Fourier series?

5 October, 2013 at 5:44 pm

Terence Tao

For continuously differentiable functions, I believe that one has conditional uniform convergence, but not absolute convergence even at a pointwise level, let alone uniformly. (A random Fourier series $\sum_{n=1}^\infty \pm \frac{1}{n} \cos(2\pi n x)$ will almost surely give a counterexample for the latter.)

5 October, 2013 at 7:20 pm

Jack

I don’t understand what you mean by “conditional uniform convergence”. There is a remark in Stein and Shakarchi’s Fourier Analysis which says that the Fourier series of $f$ converges absolutely, assuming only that $f$ converges absolutely (and hence uniformly to $f$ ) if $f$ satisfies a holder condition of order $\alpha>1/2$ . Where do I get things wrong?

5 October, 2013 at 7:36 pm

Terence Tao

Ah, you’re right; continuous differentiability (or Holder continuity with exponent greater than 1/2) does indeed give absolute uniform convergence and not just conditional uniform convergence. (I had misplaced a factor of $\sqrt{n}$ in my previous calculation.)

14 December, 2013 at 5:40 pm

Dear Prof Tao,
I have some problem for exercise 18.4.3,how I get the result frome 18.4.2?
Thank you!

23 February, 2014 at 3:45 am

Martin Scorsese

Dear Prof. Tao,

is there some reason you define elements of $\mathbb{R}^n$ to be row vectors instead of column vectors ? In every linear algebra book that I looked (e.g. Langs “Linear Algebra” oder Axlers “Linear Algebra”) and also in analysis books, like Rudins “Principles of Mathematical Analysis”, these are (in the case of the analysis books usually only implicitly) treated as column vectors (since no transposition is involved when multipliying them with matrices). Therefore I think there have to be other good reasons, which are unkown to me, to treat them as row vector if one is thereby accepting the additional hassle of having to transpose them before multiplying them against matrices…

Also regarding linear algebra you wrote on the 17 April, 2013 at 9:07 am, that there is a canonical isomorphism that maps every number $L$ to the map $x\mapsto Lx$. Why is this map “canonical” ? For one-dimensional vector spaces $V$ the correspondence $V\leftrightarrow V^*$ isn’t canonical since it depends on a chosen basis of $V$. Therefore also the correspondence $\mathbb{R}\leftrightarrow \mathbb{R}^*$ which you mentioned doesn’t seem to be canonical.

23 February, 2014 at 8:39 am

Terence Tao

Unfortunately, the most natural conventions in linear algebra have evolved to be the opposite of the conventions for the rest of mathematics. For instance, elements of a Cartesian product $X \times Y$ are invariably written as horizontal ordered pairs $(x,y)$ rather than vertical ones $\begin{pmatrix} x \\ y \end{pmatrix}$ ; similarly, functions $f$ of two variables $x,y$ are denoted $f(x,y)$ rather than $f \begin{pmatrix} x \\ y \end{pmatrix}$ , and so forth. This is probably for historical typesetting reasons rather than for any mathematical reason (and, as one can see here, column vectors still waste a lot of whitespace in modern typesetting), but still, if one insists on ${\bf R}^n$ as column vectors, then basic statements such as ${\bf R}^{n+m} = {\bf R}^n \times {\bf R}^m$ require some modification. So the compromise has been to use column vectors in situations in which linear algebra is the dominant focus, but to use row vectors if it is not (and to convert between the two by transposes if the focus varies within the text).

Ultimately, the source of the conflict is the fact that we do not use the “reverse Polish” notation of putting operations after their inputs, instead of before their inputs, e.g. $(x,y)f$ instead of $f(x,y)$ or $f \begin{pmatrix} x \\ y \end{pmatrix}$ , but it is far too much trouble to change over the entire corpus of mathematical writing (and habit) for such minor notational optimisations, as the “repaired” notation just looks too “wrong” to almost all practising mathematicians, including myself. (Though there are certainly exceptions: for instance, many mathematicians who work with group actions have a preference for right actions over left actions.)

As for your second question, ${\bf R}$ does have a canonical basis, namely the multiplicative unit 1, and hence a canonical identification with its dual. Here we are implicitly viewing ${\bf R}$ not just as a one-dimensional vector space, but as the much richer structure endowed with the structure of a field (and with many other structures besides: order structure, topological structure, measure structure, etc.).

27 February, 2014 at 2:13 am

Scorsese

I’m really sorry I have to bother you again, but I thought hard about your explanation and I can’t seem to understand these two arguments of yours:
$\bullet$ why statements like $\mathbb{R}^{m+n}=\mathbb{R}^m \times \mathbb{R}^n$ require some modifications and
$\bullet$ why reverse Polish notation would solve this conflict.

These are my concerns:
Regarding the first problem, set-theoretically $\mathbb{R}^{m+n}=\mathbb{R}^m \times \mathbb{R}^n$ , these actually means that between the set of tuples of the form $(x_1,\ldots,x_{m+n})$ and the set of tuples of the form $((x_1,\ldots,x_{m}),(x_{m+1},\ldots,x_{m+n})$ exists an (obvious) bijection – but, as you wrote at the top of pp. 73 – we neglect these minor distinctions and therefor write $\mathbb{R}^{m+n}=\mathbb{R}^m \times \mathbb{R}^n$ as if the sets were indeed the same.
But writing columnvectors instead of row vectors doesn’t change anything since then – completely analoguous to the row-vector case – we would again have a bijection

$\begin{pmatrix} x_1 \\ \vdots \\ x_{m+1} \\ \vdots \\ x_{m+n} \end{pmatrix} \mapsto \begin{pmatrix} \begin{pmatrix} x_1 \\ \vdots \\ x_m \end{pmatrix} \\ \begin{pmatrix} x_{m+1} \\ \vdots \\ x_{m+n} \end{pmatrix} \end{pmatrix}$
and could choose to neglect this minor difference.

Regarding the second problem: Polish notation is just a notation whereas row and column vector are objects that have a mathematical existance (since e.g. for column vectors of length 2 multiplication from the left with a $2\times 2$ matrix is defined but for row vector it isn’t). So how can a typographical change induce a mathematical change ?

27 February, 2014 at 6:11 am

Terence Tao

For the first point: one can certainly consistently use column notation for both vectors and ordered pairs as you do here, but the problem is typographical: due to the way English is written (from left to right, and then top to bottom), using vertical notation like this is very wasteful in space, does not look aesthetically appealing, and is also more cumbersome to write in formatting languages such as LaTeX. Furthermore, outside of linear algebra, ordered pairs (and n-tuples) are invariably written in row notation instead of column notation, so switching everything over to columns is a much larger change to mathematical notation than switching everything over to rows. (In an alternate universe in which the dominant language was written from top to bottom, rather than from left to right, as is the case for instance with traditional Chinese, it would make sense to use column notation throughout, but this is not the case in the universe we actually live in.)

For the second point: when one switches to reverse Polish notation, the natural association between linear operators and matrices changes. In orthodox notation, a linear transformation $T: {\bf R}^n \to {\bf R}^m$ between standard Euclidean spaces is associated to an $m \times n$ matrix $A$ by the formula $Tx = Ax$ for all $x \in {\bf R}^n$ , where elements of ${\bf R}^n$ are viewed as column vectors. In reverse Polish notation, it is instead natural to associate a linear transformation $T: {\bf R}^n \to {\bf R}^m$ to an $n \times m$ matrix $A$ by the formula $xT = xA$ for all $x \in {\bf R}^n$ , where elements of ${\bf R}^n$ are now viewed as row vectors. Note also that the dimensions of the matrix are now ordered in the same way as we order the domain and range of the function, i.e. with the domain dimension $n$ to the left of the range dimension $m$ . This is related to how the operation of applying a function $f$ and then a function $g$ becomes $fg$ or $f \circ g$ (or $f g \circ$ ) in reverse Polish notation, as opposed to $gf$ or $g \circ f$ in orthodox notation: reverse Polish notation makes function composition consistent with the visual operation of concatenation of arrows. (In orthodox notation, one can “repair” this issue by reversing the arrows and placing the domain to the right of the range, e.g. $f: Y \leftarrow X$ (or $Y \leftarrow X: f$ ) instead of $f: X \to Y$ , but this usually deviates too far from orthodox practice to be widely used.)

28 February, 2014 at 1:28 pm

Scorsese

You are great! Thank you so much!

10 March, 2014 at 2:15 pm

Jack

As I run through the book, I don’t see the topic “Differentiation under the integral sign“.

In the preface, you point out that careless interchanging integrals (limits, limits and derivatives, derivatives, limits and integrals) might lead to big mistakes. You’ve also talked about “There’s more to mathematics than rigour and proofs“.

I’ve seen two of your blog posts mentioned in the beginning that we can ignore the issue of “differentiation under the integral sign” for the sake of “formal level of analysis”:

“Noether’s theorem, and the conservation laws for the Euler equations“:

Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign

“Some notes on Bakry-Emery theory“:

we will ignore all technical issues of regularity and decay, and always assume that the solutions to equations such as (1) have all the regularity and decay in order to justify all formal operations such as the chain rule, integration by parts, or differentiation under the integral sign.

You once said that in the “post-rigorous” stage (of one’s mathematical education) when the emphasis is on applications, intuition, and the “big picture”, one should be able to convert all (formal) calculations into a rigorous argument whenever required.

Is there a “simple” rule of thumb that one can justify “differentiation under the integral sign”? (It seems that this is very “easy” according to those two blog posts.) To what extend can one safely do this sort of formal discussion?

As I see from lots of (should I say “all”?) your advanced discussions in analysis, almost none of the exact statement in the book (except some really big theorem like Weierstrass’s Theorem) would be mentioned. How should one understand this phenomenon?

10 March, 2014 at 3:24 pm

Terence Tao

Writing a derivative as a limit of Newton quotients, the problem of interchanging derivatives and integrals is more or less equivalent to a problem of interchanging limits and integrals. So, for instance, if the Newton quotients are all uniformly dominated by an absolutely integrable function, the dominated convergence theorem will allow one to justify the interchange. (This is for instance the case if the functions involved are smooth and compactly supported in all parameters; one can weaken these hypotheses further of course.)

As for your last question, the short answer is that most of modern analysis is not so much about theorems, but rather about techniques and ideas; see for instance Tim Gowers’ “The two cultures of mathematics” for further discussion of this point.

24 March, 2014 at 12:00 pm

Anonymous

Terence,

I am currently an undegraduate who is going to do Real Analysis next semester. While Rudin is the most used book, several reports claim that it is not “user-friendly”, whatever that means in a mathematical context. In case I feel the same way; does your books on analysis serve as an alternative? In particular, are your books suitable for Real Analysis? (I am not sure what to take from “Honors Analysis”, as honors-classes are not practiced in Norway.)

24 June, 2014 at 5:56 am

Martin Scorsese

Dear Prof. Tao,

I’m don’t really understand the point of most of the exercises (for example exercise 15.6.11 or 15.6.13) on complex numbers from pp. 502 (in the first edition).
These exercises mostly concern purely topological notions of $\mathbb{C}$ , which, by the set-theoretic definition of $\mathbb{C}$ as $\mathbb{R}^2$ together with a suitable multiplication, have already been established and proved earlier in the text.
Therefore the proof of these exercises is just a “proof by notation” and their proof has an somewhat disturbing, “tautological” feeling to it – also because they are phrased in a way as if we were proving a new result.
For example in 15.6.11 $f$ is just the identity, written in a fancy way as $f(a,b)=(a,b)=(a,0)+(0,1)\cdot (0,b)=a+ib$ using complex multiplication, and thus trivially a homeomorphism and in 15.6.13 it seems to me one doesn’t really need 15.6.11, since set-theoretically $\mathbb{C}=\mathbb{R}^2$ and thus the proof of the Heine-Borel theorem from earlier on can be used verbatim.
I’m sure you had your reasons, but wouldn’t a general remark, that the topological aspects of $\mathbb{C}$ are already all established because they have been proved earlier for $\mathbb{R}^2$ , have been sufficient ?

24 June, 2014 at 8:55 am

Terence Tao

Hmm, this is a good point. I guess the issue was blurred somewhat by overloading the $(a,b)$ notation to represent both the ordered pairs used in the Cartesian product, and the formal version of $a+bi$ in the definition of the complex numbers; it’s slightly better conceptually to think of ${\bf R}^2$ as a model (or as a coordinate chart) of ${\bf C}$ , rather than being set-theoretically identical to ${\bf C}$ . The distinction between ${\bf C}$ and ${\bf R}^2$ can be important if one wishes to consider other ways to coordinatise the complex plane, or if one wishes to exploit general linear symmetries on the plane ${\bf R}^2$ that do not preserve the complex structure. Unfortunately I did not set up the notation properly to emphasise this distinction (to do so, I would have used something like $a \oplus b \textbf{i}$ rather than $(a,b)$ to represent a formal complex number, although that looks somewhat ugly and perhaps would have caused additional confusion).

(There is also one tiny piece of mathematical content to Exercise 15.6.11, which is to check that the Euclidean metric on ${\bf R}^2$ agrees with the complex metric on ${\bf C}$ , although this is trivially verifiable in one line, of course.)

These sorts of questions become more relevant if one is studying more complicated spaces, e.g. the Riemann sphere ${\bf CP}^1$ , which has two coordinate charts by ${\bf C}$ (or by ${\bf R}^2$ ), in which case establishing basic topological properties of this space (e.g. completeness or connectedness) requires slightly more effort, but I think I agree with you that in this context a remark that ${\bf C}$ is identical as a metric space to ${\bf R}^2$ is simpler. (This does reinforce the slightly incorrect notion that ${\bf C}$ “is” ${\bf R}^2$ as a mathematical space, not just as a metric space, but in the grand scheme of things this is a fairly harmless notion to hold.)

26 August, 2014 at 12:03 am

Anonymous

Dear Prof. Tao, i have a confusion:
in Proposition 12.2.15 :
(d) Any singleton set {xo}, where xo E X, is automatically
closed.

but in 12.3 Relative topology, you show us a example:
however if instead
one uses the discrete metric ddisc, then { 1} is now an open set

26 August, 2014 at 10:23 pm

Terence Tao

It is possible for a set to be simultaneously open and closed (such sets are known as clopen sets).

26 August, 2014 at 11:05 pm

Anonymous

the intuitive is ingrained. i must forget those common sense. thank you for your patience.

26 August, 2014 at 10:55 pm

Anonymous

It is possible for a set to be simultaneously
open and closed, if it has no boundary. so no confusion.

3 September, 2014 at 11:16 pm

Anonymous

Exercise 12.5.8. Let ( $X, d_{l^1}$ ) be the metric space from Exercise 12.1.15.
For each natural number n, let $e^{(n)} = {{e_j}^{(n)}}_{j = 0}^{\infty}$ be the sequence in $X$
such that ${e_j}^{(n)} := 1$ when n = j and ${e_j}^{(n)} := 0$ when $n \neq j$ Show that
the set $\{e^{(n)} : n \in N\}$ is a closed and bounded subset of $X$ , but is not
compact. (This is despite the fact that (( $X, d_{l^1}$ ) is even a complete metric
space ¨C a fact which we will not prove here. The problem is that not
that $X$ is incomplete, but rather that it is ¡°infinite-dimensional¡± , in a
sense that we will not discuss here.)

in fact , i have spend several days to prove that : ( $X, d_{l^1}$ ) is a complete metric space. but i can¡¯t got it. can you please show me some clues? thank you.

25 September, 2014 at 2:36 pm

Anonymous

Look in any good book about Functionalanalysis like Yoshida or Schechter …

3 September, 2014 at 11:29 pm

Anonymous

You can adopt the method e.g. in Schechter, Principles of Functional Analysis, 1.3, Example 3 to prove this.

4 September, 2014 at 3:34 am

Anonymous

I got it, thank you.

10 September, 2014 at 11:11 pm

Anonymous

Dear Prof. Tao
in Exercise 13.3.6. Show that the addition function (x,y) $\mapsto$ x+y and the
subtraction function (x, y) $\mapsto$ x – y are uniformly continuous from $R^2$
to R, but the multiplication function (x, y) $\mapsto$ xy is not. Conclude that
if f : X $\to$ R and g : X $\to$ R are uniformly continuous functions on
a metric space (X, d), then f + g : X $\to$ Rand f – g : X $\to$ Rare
also uniformly continuous. Give an example to show that fg: X $\to$ R
need not be uniformly continuous. What is the situation for max(f, g),
min(f,g), f/g, and cf for a real number c?

I want to proof max(f,g) is uniformly continuous.
in fact , for any $\epsilon/2$ , there exist $\delta$ , when d(x,x’) < $\delta$ , $ d(f(x),f(x')) < \epsilon/2$ and $ d(g(x),g(x')) < \epsilon/2$.

in situation max(f,g), when d(x,x') = g(x) and f(x’) >= g(x’) (or g(x) >= f(x) and g(x’) >= f(x’)) then max(f,g) = f, it’s obvious.

however, it f(x) > g(x) and g(x’) > f(x’) (or g(x) > f(x) and f(x’) > g(x’)) , i want to found one point $x_0$ , such that $f(x_0) = g(x_0)$ , $d(x,x_0) < \delta$ and $d(x',x_0) < \delta$ ,then based on Triangle inequality ,we got it.

the key point is How to found that $x_0$ ?, there seems NOT have something similar to Intermediate value theorem which need connected.

OR max(f,g) is NOT uniformly continuous?

11 September, 2014 at 1:05 am

Anonymous

i think this should work:

In fact , for any $\varepsilon$ , there exist $\delta$ , when $d_x(x,x') < \delta$ , we have: $d(f(x),f(x')) = |f(x) - f(x')| < \varepsilon$ and $d(g(x),g(x')) = |g(x) - g(x')| < \varepsilon$ .

In situation max(f,g):
Case1: $f(x) \ge g(x)$ and $f(x') \ge g(x')$ ( $g(x) \ge f(x)$ and $g(x') \ge f(x')$ is similar ) then $d(max(x),max(x')) = d(f(x),f(x')) g(x)$ and $g(x') > f(x')$ ( $g(x) > f(x)$ and $f(x') > g(x')$ is similar). based on inequality formulas on R, we have : $f(x) - f(x') > f(x) - g(x') > g(x) - g(x')$ , with $d(f(x),f(x')) = |f(x) - f(x')| < \varepsilon$ and $d(g(x),g(x')) = |g(x) - g(x')| < \varepsilon$ . we have $d(max(x),max(x')) = d(f(x),g(x')) = |f(x) - g(x')| < \varepsilon$ .

so max(f,g) is uniformly continuous.

18 September, 2014 at 7:26 pm

Anonymous

Dear Prof. Tao, in Exercise 13.5.5. Given any totally ordered set $X$ with order relation $\leq$ ,
declare a set $V \subset X$ to be open if for every $x \in V$ there exist $a, b \in X$
such that the “interval” $\{y \in X : a < y < b\}$ contains $x$ and is contained
in $V$ . Let $\mathcal{F}$ be the set of all open subsets of X. Show that $(X,\mathcal{F})$ is
a topology.

I have a problem with this exercise:

We know $X \not \subset X$ ,so $X \notin \mathcal{F}$ , and so $(X,\mathcal{F})$ is NOT a topology? ,so $V \subset X$ shoulde be $V \subseteq X$ ?

even with $V \subseteq X$ hold, later for $R ^{*}$ case , then $+\infty \in R ^{*}$ ,BUT, there DOES NOT exist $a, b \in R ^{*}$
such that the "interval" $\{y \in R ^{*} : a < y < b\}$ contains $+\infty$ , $R ^{*} \notin \mathcal{F}$ , so $(R ^{*},\mathcal{F})$ is NOT a topology?

19 September, 2014 at 8:10 am

Terence Tao

Thanks for the correction. (As for the second issue, see the errata for this question for the second edition, which has been corrected in the third edition.)

23 September, 2014 at 8:34 am

ua46f

Dear Prof. Tao, in your new Exercise 13.5.8 : Show that there exists an uncountable well-ordered set $\omega_1+1$ that has a maximal element $\infty$ , and such that the initial segments $\{ x \in \omega_1+1: x < y \}$ are countable for all $y \in \omega_1+1 \backslash \{\infty\}$ . If we give $\omega_1+1$ the order topology (Exercise 13.5.5), show that $\omega_1+1$ is compact; however, show that not every sequence has a convergent subsequence.

After follow your hint , we can get well order set : $\omega_1+1$ ,apply Transfinite induction to $\omega_1+1$ , we can get: $\{ x \in \omega_1+1: x < y \}$ are countable for all $y \in \omega_1+1 \backslash \{\infty\}$ , so a initial segment it is.

my problem is whether the condition "countable" is necessary for $\omega_1+1$ to be compact?

i have this proof which don't use the "countable" condition:

let $\omega_1+1 = \bigcup _ {c \in I} V_c$ , $I$ is some infinite set. then $\infty \in \omega_1+1$ ,there must exist some $V_{c^m}$ which contain $\infty$ , we can found $V_{c^{m-1}}$ which have : $min(V_{c^{m-1}}) < min(V_{c^{m}})$ and for every $s, min(V_{c^{m-1}}) < s < min(V_{c^{m}}) , s \in V_{c^{m-1}}$ , and so forth, then $min(V_{c^{m}}), min(V_{c^{m-1}}), \dots$ is a strick descendant sequence ,so which must be finite. so $\omega_1+1 = \bigcup _ {0 \leq n \leq m ; c_n \in I} V_{c^n}$ , a finite cover.

24 September, 2014 at 11:32 am

Terence Tao

Yes, one can replace $\omega_1$ by any other successor ordinal and still obtain topological compactness in the order topology. (Whether sequential compactness holds in that case is another question, though.)

25 September, 2014 at 6:24 am

ua46f

Dear Prof. Tao, in Exercise 13.5.13. Generalize Corollary 12.5.9 to compact sets in a topological space.

Corollary 12.5.9. Let $(X,\;d)$ be a metric space, and let $K_1,\;K_2,\;K_3$
be a sequence of non-empty compact subsets of $X$ such that
$K_1 \; \supset \;K_2 \; \supset K_3\; \supset \; \dots$
Then the intersection $\bigcap _{n=1} ^{\infty} K_n$ is non-empty.

I think we needs to assume as an additional hypothesis that the topological space is Hausdorff.

If we don’t have this hypothesis, i just don’t know how to prove $K_n$ is closed.

[Correction added, thanks -T.]

25 September, 2014 at 2:32 pm

Anonymous

Metrische Spaceshuttle are Hausdorff spaces.

25 September, 2014 at 2:34 pm

Anonymous

Sorry, metric spaces are ….

28 September, 2014 at 5:26 pm

José Antonio Lara Benítez

Dear Prof. Tao I think there is a typo in the proposition 18.3.3 instead of $m^{*}(\sum _{q\in J}q+E )= \sum _{q\in J}m^{*}(q+E)$ should be $m^{*}(\bigcup_{q\in J}q+E )= \sum _{q\in J}m^{*}(q+E)$ . In the same way at the end of the proof should be $\bigcup_{q\in J}q+E$ is a subset of $X$

[Correction added, thanks -T.]

1 November, 2014 at 2:06 pm

José Antonio Lara Benítez

Dear Professor Tao. In the Corollary 18.4.7 p. 592 I think is necessary to assume that $m(A)<\infty$ in order to avoid some indeterminate form.

[Correction added, thanks – T.]

3 November, 2014 at 5:13 pm

José Antonio Lara Benítez

Dear professor Tao. I think there is a problem with the hint in the exercise 19.1.3 p. 608. Because as the $f_n$ ‘s are defined doesn’t necessarily implies that are simple as is required in the lemma 19.1.5 p. 603.

3 November, 2014 at 11:03 pm

Terence Tao

Actually, I believe the $f_n$ defined in the hint are indeed simple (they can only take finitely many values, namely the multiples of $2^{-n}$ between $0$ and $2^n$ ).

4 November, 2014 at 12:25 am

Anonymous

in Lemma 16.4.6 , $\sum_{n=0}^{N-1}e_n(x) = \frac{e_N - e_0}{e_1 - e_0} = \frac{e^{\pi i(N-1)x} sin(\pi Nx)}{sin(\pi x)}$ then $F_N(x) = \frac{{sin(\pi Nx)}^2}{N{sin(\pi x)}^2}$

SHOULD BE $\sum_{n=0}^{N-1}e_n(x) = \frac{e_N - e_0}{e_1 - e_0} = \frac{e^{\frac{\pi i (N-1)x}{2}} sin(\frac{\pi N x}{2})}{sin(\frac{\pi x}{2})}$ then $F_N(x) = \frac{{sin(\frac{\pi Nx}{2})}^2}{N{sin(\frac{\pi x}{2})}^2}$ ?

4 November, 2014 at 12:09 pm

Terence Tao

No; note that $e_n(x)$ is equal to $e^{2\pi i nx}$ , not $e^{\pi i nx}$ .

4 November, 2014 at 5:33 pm

Anonymous

yes, i miss the definition . thank you for your time.

4 November, 2014 at 1:16 am

Anonymous

one more question: if this is case ,the condition $0 < \delta < 1/2$ , $\delta \leq |x| \leq 1 - \delta$ SHOULD BE $0 < \delta < 1$ , $\delta \leq |x| \leq 1$ this is same to Definition 14.8.6

4 September, 2015 at 3:39 am

Xianjin

Dear Professor,

for the proof of theorem 19.2.9 Lebesgue monotone convergent theorem, why do we need $(1-\epsilon)$ ? Can we change the proof like this? $\forall x \in \Omega s(x) \leq \sup f_k (x)$ , then $\forall x in \Omega, \exists N_x, \forall k \geq N_x, f_k (x) \geq s(x)$ . Define $E_k$ as $E_k := {x \in \Omega: f_k(x) \geq s(x)}$ , then following the same reasoning as the end of the proof in the book. Is this still correct? Thanks!

xianjin

4 September, 2015 at 7:11 am

Terence Tao

$s(x) \leq \sup f_k(x)$ does not imply that $f_k(x) \geq s(x)$ for all sufficently large $k$ (e.g. consider the example $s(x)=1$ , $f_k(x) = 1-1/k$ ). One would need $s(x) < \sup f_k(x)$ for this (which is more or less what the $\varepsilon$ is achieving).

20 November, 2015 at 4:37 am

pingping

Dear Professor Tao
I’m a physics sophomore student of ordinary Chinese university. Now I’ve read the book “analysis” and “linear algebra done right”.because I’m majoring at physics,to be suggested,I don’t need the math very rigorously.I love it,and I benefit from it a little which it don’t pay equally though.I want to a university teacher or tutor who can study physics by deducing formula and theorem combination experimental data in my free time.If fortunately,I maybe a theoretical physicist.
So this is first question.
1.If we need rigorous exercise in the real analysis and linear algebra,when we don’t need in the as rigorous as exercise advance ?(for math and for physics).
2.I want to learn rigorous math knowledge which used in physics.Do it make sense for my dream?
3.There rarely is book like you written.So,what’s your suggestion about complex analysis?what about the book written by pro Elias M.Stein?
4.I don’t know what should I learn next,can you give me some idea about the knowledge I need or some book.I try my best to learn the knowledge which the physics need.
So,could you tell me something,if ok ,I will be very appreciate it.
Thank you very much! Please excuse my english.
亲爱的陶教授
由于我的英文有限，我用中文再表达一下我的想法。您启蒙了我的数学思想，这正是我想的在大学里学的数学，我因为您的书爱上了数学。但是我学数学是更好的服务于物理。我想以后无论做什么工作，我都可以在闲暇时间研究研究物理，听听物理前沿的seminar。我更希望成为我的职业，我希望可以做一名既可以解析推导物理，也可以结合实验数据研究物理的学者。但是由于像您这样严格的书确实很少，学习严格的数学知识几乎占用了我的所有业余时间，所以想让您指点迷津，给我一些建议，推荐一些书籍。如果可以的话，我会很感激您的教导，谢谢！

16 December, 2015 at 8:35 pm

pingping

Dear Professor Tao
I’m a physics sophomore student of ordinary Chinese university. Now I’ve read the book “analysis” and “linear algebra done right”.because I’m majoring at physics,to be suggested,I don’t need the math very rigorously.I love it,but I benefit from it a little which it don’t pay equally though.I want to be a university teacher or tutor who can study physics by deducing formula and theorem combination experimental data in my free time.If fortunately,I maybe a theoretical physicist.
So this is the first question.
1.If we need rigorous exercise in the real analysis and linear algebra,when we don’t need in the as rigorous as exercise in advance curriculum ?(for math and for physics).
2.I want to learn rigorous math knowledge which used in physics.Do it make sense for my dream?
3.There rarely is book like you written.So,what’s your suggestion about complex analysis?what about the book written by pro Elias M.Stein.?If it is worth to be study for a physics student?
4.I don’t know what should I learn next,can you give me some idea about the knowledge I need or some book.I try my best to learn the knowledge which the physics need.
So,could you tell me something,if ok ,I will be very appreciate it.
Thank you very much! Please excuse my english.
亲爱的陶教授
由于我的英文有限，我用中文再表达一下我的想法。您启蒙了我的数学思想，这正是我想的在大学里学的数学，我因为您的书爱上了数学。但是我学数学是更好的服务于物理。我想以后无论做什么工作，我都可以在闲暇时间研究研究物理，听听物理前沿的seminar。我更希望成为我的职业，我希望可以做一名既可以解析推导物理，也可以结合实验数据研究物理的学者。但是由于像您这样严格的书确实很少，学习严格的数学知识几乎占用了我的所有业余时间，所以想让您指点迷津，给我一些建议，推荐一些书籍。如果可以的话，我会很感激您的教导，谢谢！

6 March, 2016 at 1:14 am

Zelin

Dear Professor Tao,

Proposition 1.5.5 in Analysis II seems a bit confusing to me. It says

Let $\left( X,d \right)$ be a compact metric space. Then $\left( X,d \right)$ is both complete and bounded.

But there is no definition of a bounded metric space before this proposition. Is it proper to formulate as follows?

Let $\left( X,d \right)$ be a compact metric space. Then $\left( X,d \right)$ is complete and $X$ is bounded.

7 March, 2016 at 12:10 pm

Terence Tao

One can add to Definition 1.5.3 that a metric space $(X,d)$ is called bounded if $X$ is bounded.

9 March, 2016 at 3:54 am

Zelin

Dear Professor Tao

The part b) of the Corollary 2.2.3 in Analysis II says that

If $f$ and $g$ are continuous, $g\left( x \right) \neq 0$ for all $x \in X$ , then $f/g: X \rightarrow \mathbb{R}$ is also continuous at $x_{0}$ .

I’m afraid their was a typo here. It might probably be

…, then $f/g: X \rightarrow \mathbb{R}$ is also continuous.

[Correction added, thanks – T.]

27 March, 2016 at 12:02 pm

Anonymous

Consider a smooth map $f:{\bf R}^m\to{\bf R}^n$ and let $A=(a_{ij})$ with

$\displaystyle a_{ij}=\frac{\partial f_i}{\partial x_j}$ .

When $m\neq n$ , it is easy to tell whether the Jacobian should be $A$ or the transpose $A^t$ , depending on one uses column or row vectors.

When $m= n$ , how can one tell quickly that which one ( $A$ or $A^t$ ) is correct?

27 March, 2016 at 12:07 pm

Anonymous

I should have make my point clearer. Let $f:{\bf R}^m\to{\bf R}^n$ be a smooth map. by definition, $Df(x)$ gives a linear transformation at $x$ from ${\bf R}^m$ to ${\bf R}^n$ . When $m=n$ , both $A$ and $A^t$ give a linear transformation from from ${\bf R}^m$ to ${\bf R}^n$ . How can one tell which one is wrong when, say, the column vectors are used?

3 April, 2016 at 5:39 am

Anonymous

The conditions for the Theorem 17.7.2 (inverse function theorem) fail for the following example
$\displaystyle f(x,y)=(x^2y+x,6x+y^2)$
at $(1,1)$ .
Can one thus argue that $f$ does not have a local inverse at $(1,1)$ ?

3 April, 2016 at 9:54 am

Terence Tao

No. For instance, $f(x) = x^3$ does not obey the hypotheses of the inverse function theorem at $x=0$ , but is still locally invertible.

In your specific example, it looks like $f$ does indeed have a singularity (probably a fold singularity) at $(1,1)$ that makes it non-invertible, but this requires a certain amount of calculation to verify (basically one has to “blow up” or “resolve” the singularity by repeated change of variables until one can see the structure of the singularity). When the degeneracy of the map is of fairly low degree, one can do this by ad hoc methods, but for high degree singularities it is best to proceed by the general machinery of algebraic geometry.

13 April, 2017 at 11:33 am

Anonymous

Do you have a reference for the comment that “basically one has to ‘blow up’ or ‘resolve’ the singularity by repeated change of variables until one can see the structure of the singularity”? I’m very much interested in how this is done in details.

What “the general machinery of algebraic geometry” are you referring to in the comment above?

[See https://en.wikipedia.org/wiki/Resolution_of_singularities – T.]

13 June, 2017 at 1:36 pm

Anonymous

Can we use Theorem 2 (Brouwer invariance of domain theorem) discussed in this post:

https://terrytao.wordpress.com/2011/06/13/brouwers-fixed-point-and-invariance-of-domain-theorems-and-hilberts-fifth-problem/

for the case $f(x,y)=(x^2y+x,6x+y^2)$
at $(1,1)$ ?

14 September, 2016 at 3:17 am

Zelin

Dear Professor Tao,

The Corollary 3.8.18 and Corollary 3.8.19 both have

Let $f: [0,1] \to \mathbb{R}$ be a continuous function supported on $[0,1]$ , …

which is a bit confusing. I guess the function does not have to be supported, does it?

PS: I happened to see that the Analysis series are going to be published by Springer, which is a great news since I am a big fan of Springer. Will the corrections up to now be included in the new edition?

[Corrected, thanks. There are currently no plans for a fourth edition. -T.]

2 October, 2016 at 12:00 am

Repository Alpha

For those who are not aware Analysis II has been published in its third edition at Springer, https://www.springer.com/gb/book/9789811018046

@Terry What are all the differences between this edition and the second one? Thanks!

4 October, 2016 at 5:44 am

Anonymous

Dear Prof. Tao,

in the proof of Lemma 6.6.6 you write that “the same argument shows that for sufficiently small $\varepsilon>0$ , $F$ maps the closed ball $\overline{B(0,r-\varepsilon)}$ to itself.”
Can you explain why this is the case? For me it looks like we just get the upper bound $||F(x)||< r-\frac{\varepsilon}{2}$ for any $x\in \overline{B(0,r-\varepsilon)}$ by using the same argument. Am I missing something?

[For $\varepsilon$ small enough, one has $\|y\| \leq \frac{r}{2} - \varepsilon$ . -T.]

Furthermore, could you give me a hint for exercise 6.6.8? Following a similar approach as in the proof of the contraction mapping theorem I am only able to establish an upper bound for $d(x_0,y_0)$ which is much bigger than the one mentioned in the exercise.

[Find as many inequalities as possible relating $d(x_0,g(x_0))$ , $d(g(x_0),y_0)$ , and $d(x_0,y_0)$ with each other and with $\varepsilon$ to get the bound $d(x_0,y_0) \leq \varepsilon / (1 - c')$ . -T.]

19 December, 2016 at 4:50 am

Anonymous

In this book, a rigorous definition given by power series for the function $\sin x$ . One can then easily prove the limit

$\displaystyle \lim_{x\to 0}\frac{\sin x}{x}=1. \qquad (1)$

Starting from the very beginning of Chapter I in Analysis I, one builds up step by step upon the Peano axioms, and eventually gets to the proof of (1).

Having a reflection on the undergraduate calculus, I’m trying to justify the “proof” for (1) in the common used calculus textbook (by Stewart), which essentially uses a geometric argument with the notions of “arc length” and “angles”.

In order to use these two notions rigorously and the corresponding geometric definition of $\sin x$ , does one need more axioms than the presented ones in Analysis I and II? Or are they hinted somewhere already?

19 December, 2016 at 9:24 am

Terence Tao

To make the geometric proof of (1) fully rigorous, one needs a modern rigorous treatment of Euclidean geometry, including the identification of the Euclidean plane with the Cartesian plane ${\bf R}^2$ . This is not done in my text, but can be found in many modern Euclidean geometry texts. Euclid’s original axioms are not quite suitable for this task (Euclid being, of course, unaware of the modern framework of first-order logic and set theory that now forms the standard foundation of mathematics), but one can use for instance Hilbert’s axioms for this purpose. Also, one eventually has to show that the geometric definition of $\sin(x)$ (from trigonometry) and the analytic definition (from power series) are equivalent. This can be done for instance by verifying that both definitions solve the same ordinary differential equation with the same initial conditions, and appealing to the Picard uniqueness theorem for such ODE; there are a number of other proofs as well (e.g. I sketch a complex analytic proof in these lecture notes).

3 January, 2017 at 3:30 am

Anonymous

Dear Prof. Tao,
Your book is very well-written, rigorous and clear. You may consider in the next edition discussing the change of variables formula in the last chapter. This will be very helpful to students as the change of variables formula is always assumed in books on differential topology and differential geometry. And if possible it would also be good to add a new chapter about integration of differential forms and stokes’ theorem.

3 January, 2017 at 3:38 am

Anonymous

And if possible please combine the two volume into one. A book with 600-700 pages is totally acceptable. And often the second volume refers to statements in the first volume, it is slightly inconvenient to put down the book and pick up another book to check and return…..

13 February, 2017 at 5:54 am

Anonymous

Is it true that for every sequence of real numbers $a_n$ there exists a function whose n-th derivative at zero is equal to $a_n$ for all $n$ ?

Due to the existence of power series whose radius of convergence is $0$ , would that contradict the Taylor’s theorem?

27 June, 2017 at 9:16 am

lorenzo

typo: on page 18 (numbering of the third edition) after Lemma 1.4.3, “Proof. See Exercise 1.4.3” should be “Proof. See Exercise 1.4.1”

[Corrected, thanks – T.]

21 July, 2017 at 11:41 pm

Biswaranjan Behera

1. Page 140 (third edition), last paragraph: “continuous on F” should be “continuous at $x_0$ .
2. Page 146. Exercise 6.4.1 is on proving a linear transformation to be continuously differentiable, but this concept is defined on the next page in Definition 6.5.1. (It also appears in the second paragraph on Page 146.)

[Corrections added, thanks – T.]

7 August, 2017 at 11:13 pm

Biswaranjan Behera

1. Page 159 (third edition). In the expression involving $DF(y)$ , the partial derivatives should be those of $F$ instead of $f$ .
2. Page 161, line 13. It should be “However, if some other derivative $\frac{\partial f}{\partial {x_j}}$ is not zero,”

22 August, 2017 at 7:18 am

lorenzo

In Proposition 2.3.2 (Maximum principle) (numbering of the third edition) shouldn’t we assume also that $X$ is non-empty?

23 August, 2017 at 5:11 am

lorenzo

Proposition 2.3.2 can be proved without assuming that $X$ is non-empty, don’t mind what I asked before.

19 September, 2017 at 7:07 am

lorenzo

typo: on page 48 (numbering of the third edition) in the text of Exercise 3.1.5 “…conclude that $\lim_{x\in x_0; x\in E} g\circ f(x)=z_0$ .” should be “…conclude that $\lim_{x\to x_0; x\in E} g\circ f(x)=z_0$ “.

[Correction added, thanks -T.]

28 September, 2017 at 10:09 am

lorenzo

on page 53 (numbering of the third edition), in the text of Exercise 3.2.2 (c) shouldn’t $g\colon (-1,1)\to\mathbb{R}, g(x):= x/(1-x)$ be $g\colon (-1,1)\to\mathbb{R}, g(x):= 1/(1-x)$ ? Because you state that the partial sums $\sum_{n=1}^N f^{(n)}$ converge pointwise to $g$ as $N\to\infty$ on $(-1,1)$ but if we take $0.5\in(-1,1)$ then we have $\lim_{n\to\infty} \sum_{n=1}^N f^{(n)} (0.5)=1/(1-0.5)=2\neq 1=g(0.5)$ , a contradiction.

[Correction added, thanks – T.]

16 November, 2017 at 5:37 am

lorenzo

In Exercise 4.7.1, to prove the trigonometric identities can we use the fact that $(e^{ix})^2=e^{2ix}$ ? In the preceding paragraph the book mentions that the complex exponential obeys many of the properties of the real exponential but only one (namely that $exp(z+w)=exp(z) exp(w)$ ) is mentioned.

16 November, 2017 at 5:47 am

lorenzo

Ah of course it’s true by the same property I mentioned, don’t mind what I wrote before.

11 December, 2017 at 4:06 am

lorenzo

Typos in the third edition:
on page 113, in Exercise 5.2.3 “…if $0<A\leq B$ are real numbers, so that…" should be "…if $0<A\leq B$ are real numbers, show that…";
on page 116, in Theorem 5.4.1 "Then there exists a trignometric polynomial…" should be "Then there exists a trigonometric polynomial…".

Also, I think that on page 53 the text was correct as it was since the series starts from 1 and not 0; sorry for that.

[Errata added, thanks – T.]

11 December, 2017 at 4:16 am

lorenzo

Another typo:
on page 129 in Example 6.1.8 “…defined by a clockwise rotation…” should be “…defined by a counterclockwise rotation…”

[Erratum added, thanks -T.]

11 December, 2017 at 5:29 am

lorenzo

Yet another typo:
on page 133 “…and thus we have $a_{ij}=b_{ij}$ for every $1\leq i\leq m$ and $1\leq j\leq m,...$ ” should be $a_{ij}=b_{ij}$ for every $1\leq i\leq m$ and $1\leq j\leq n,...$

[Erratum added, thanks -T.]

18 December, 2017 at 2:48 pm

lorenzo

Some typos:
– on page 141 in the last line the period at the end should be deleted since page 142 starts with a lowercase letter;
– at the end page 142 $(\sum_{j=1}^n v_{j}\frac{\partial f_i}{\partial x_j}(x_0))_{i=1}^m$ I think should be $(\sum_{j=1}^n v_{j}\frac{\partial f_i}{\partial x_j}(x_0))_{1\leq i\leq m}$ ;
– on page 144 in Exercise 6.3.2 $D_{e_j}f(x_0)=D_{-e_j}f(x_0)$ I think should be $D_{e_j}f(x_0)=-D_{-e_j}f(x_0)$ .

[Corrections added, thanks – T.]

26 December, 2017 at 7:02 am

lorenzo

– On page 56 there’s a missing parenthesis: on line (c) “(Definition 3.2.1; and” should be “(Definition 3.2.1); and”;
– In Exercise 6.6.8 on page 152 it seems to me that it should be $d(x_0,y_0)\leq\frac{\varepsilon}{1-\max(c,c')}$ (since $d(x_0,y_0)\leq\frac{\varepsilon}{1-c'}$ ).

[Corrections added, thanks – T.]

6 January, 2018 at 2:22 pm

lorenzo

typo: on page 156 “Let $(x_n)_{n=1}^{\infty}$ be any sequence in $V-0$ ” should be “Let $(x_n)_{n=1}^{\infty}$ be any sequence in $V - \{0\}$ “.

[Correction added, thanks – T.]

29 March, 2018 at 6:07 am

Xueping

On page 138(Analysis II, 3rd ed.), Example 6.3.3, I think $D_{-1}f(x)$ is minus left derivative of f(x) (if it exists).

[Corrected, thanks – T.]

17 April, 2018 at 4:08 pm

Xueping

On page 156, in the proof of inverse mapping theorem, when defining the open set U on which f is injective, possibly one need take the intersection of f^{-1}(V) with \bar{B}(0,r). A bit more explanation is required: U\cap \bar{B}(0,r) is indeed a subset of B(0,r) and is therefore open.

[Corrected, thanks – T.]

8 May, 2018 at 6:41 am

Anonymous

College students were punished by some teacher for using L’Hopital rule when calculating the limit

$\displaystyle \lim_{x\to 0}\frac{\sin x}{x}.$

The reason given was that this limit can be identified as the derivative of $f(x)=\sin x$ at $x=0$ and thus using L’Hopital rule in the calculation is circular.

Is it really so? Although L’Hopital is an unnecessary powerful gun to used here, is there a way to justify that one can use it in a non-circular way to calculate the limit above?

8 May, 2018 at 7:36 am

Terence Tao

This is highly unlikely. Note for instance that if one defined $\sin x$ using degrees instead of radians, then the limit above would equal $\frac{2\pi}{360}$ rather than $1$ . So any non-circular evaluation of this limit must, at some point, must use the fact that the sine is defined using radians instead of degrees; thus for instance one cannot hope to just use (say) the trigonometric addition formulae to resolve this limit, as they are valid for both degrees and radians.

8 May, 2018 at 8:15 am

Anonymous

I think the following is “non-circular”?

Define as what was done in your book $\sin x$ and $\cos x$ using power series. Then prove using term-by-term differentiation and convergence of power series that $(\sin x)'=\cos x$ . Now one can invoke L’Hopital’s rule to calculate the limit.

Of course, once one proves that $(\sin x)'=\cos x$ one can say that the value of that limit is $1$ IF one identifies that limit as $\lim_{x\to 0}\frac{\sin x-\sin 0}{x-0}$ and invokes the definition of derivatives. (Alternatively, one may deal with the limit directly with the power series definition without involving any derivative business.) One may say that it is rather “stupid” or unnecessary to use L’Hopital in this scenario. But maybe using L’Hopital here is not logically wrong as long as one can check all the assumptions of the theorem?

8 May, 2018 at 8:19 am

Anonymous

I should mention that in the L’Hopital-approach in the second paragraph above one needs to prove the continuity of $\cos x$ near $x=0$ as well.

8 May, 2018 at 9:48 am

Terence Tao

Sure, if one already proves the stronger statement $(\sin x)' = \cos x$ (using for instance the power series definition for both functions) then one could then use L’Hopital’s rule somewhat gratuitously to then conclude the special case $\lim_{x \to \infty} \frac{\sin x - \sin(0)}{x - 0} = \cos(0)$ of the above statement. Now the argument is non-circular, but it is the argument establishing the stronger statement $(\sin x)' = \cos x$ which is doing all of the work, L’Hopital’s rule is not actually adding anything of substance to the argument. One could similarly make a non-circular proof of any theorem X in mathematics “using L’Hopital’s rule” by first proving X without using that rule, using L’Hopital’s rule to prove some trivial statement (e.g. showing that $1=1$ by evaluating the limit $\lim_{x \to 0} \frac{x}{x}$ in two different ways), and then combining X with the trivial statement to again recover X. While this may still be a technically valid argument, if I had to grade a proof of X that proceeded in this fashion, I would also be inclined to deduct points on the grounds that it demonstrates that the student does not understand what parts of the argument are the most essential, and is wasting the reader’s time with unnecessary trivialities.

8 May, 2018 at 11:48 am

Anonymous

[Typo $x\to\infty$ in your comment:-)]

Hmm, I would be totally convinced by points deduction based your ground of “unnecessary trivialities” (especially in this special case) while I think a zero grade based on the ground of “logically incorrect” or “circular” as I have seen before would be a severe false accusation, which would a student rather frustrated. In some similar cases, it seems that L’Hopital’s rule may be more a “psychological shortcut” than an unnecessary powerful gun. For instance, in a step of evaluating the limit

$\displaystyle \lim_{x\to 0}e^{\frac{1}{x}\ln (1+\frac{x}{3})},$

it seems “natural” to write

$\displaystyle \lim_{x\to 0}\frac{\ln(1+\frac{x}{3})}{x} =\lim_{x\to 0}\frac{\frac{1}{3}\cdot \frac{1}{1+\frac{x}{3}}}{1}=\frac{1}{3}$

while it “should” be done as

$\displaystyle \lim_{x\to 0}\frac{\ln(1+\frac{x}{3})}{x} =\lim_{x\to 0}\frac{\ln(1+\frac{x}{3})-\ln(1+\frac{0}{3})}{x-0}=[\ln(1+\frac{x}{3})]'|_{x=0} =\frac{1}{3}\cdot\frac{1}{1+\frac{0}{3}}.$

It may be more “psychologically convenient” to take derivatives on both the numerator and denominator of the fraction $\frac{\ln(1+\frac{x}{3})}{x}$ than to identify the fraction $\frac{\ln(1+\frac{x}{3})}{x}$ as a difference quotient $\frac{\ln(1+\frac{x}{3})-\ln(1+\frac{0}{3})}{x-0}$ , the limit of which is the definition of some derivative.

Anyway, it seems that the process of proving certain result X using the established results is quite different from calculating some Y with all the powerful guns on hand. It may be preferable (sometimes?) to perform some calculation in some unnecessarily redundant way, though it should be always instructive to identify the truly essential steps in a calculation, according to your advice (https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/).

23 August, 2018 at 12:17 am

Aditya Ghosh

In your book Analysis II (3rd Ed.), page 140, there is one more correction: In the beginning of the proof of theorem 6.3.8, $L(v_j)_{1\le j\le m}$ should be corrected to $L(v_j)_{1\le j\le n}$ and similarly the sum on the RHS should be from 1 to n (instead of 1 to m).

[Correction added, thanks – T.]

15 October, 2018 at 11:19 am

SkysubO

Dear Professor Tao,

When X is empty, is that a counterexample to Proposition 12.5.5： “Let (X,d) be a compact metric space. Then (X,d) is both complete and bounded.” ？

In my view, (Ø,d) is not bounded according to definition 12.5.3.

Or am I wrong in understanding the definition of “Bounded sets” ?

Best regards,
Su

16 October, 2018 at 10:41 am

SkysubO

Forgot to say, Analysis II，first edition，page 413.

Best regards, Su

16 October, 2018 at 11:49 pm

Anyone

Although I am not Professor Tao, I will try to answer. The empty set is bounded, since it is a subset of any other set, in particular, of any ball in the ambient space.

17 October, 2018 at 8:22 am

SkysubO

I agree with you when the ambient space X is nonempty.
When Ø is in (Ø,d), I think Ø is not bounded since there is no ball in metric space (Ø,d).

Sorry for my poor English and I hope you can understand me.

17 October, 2018 at 12:19 pm

Anonymous

There is no need to define the concept of boundedness for the (trivial) “empty metric space”.

24 October, 2018 at 7:50 am

Terence Tao

Oops, the definition of bounded set needs to be adjusted to make the empty metric space bounded. One can do this by inserting “for every $x \in X$ ” before “there exists a ball $B(x,r)$ ” in the definition.

23 October, 2018 at 10:57 pm

anonymous

Analysis II ed 3 electronic : p189 paragraph above the Remark 8.1.8 there is a full stop outside parentheses despite a bracketed full sentence; same for p82 outside of a bracket in exercise 4.2.3 whereas in your exercise 2.2.2 p32 the full stop seems printed correctly and hence different hence a contrast, and may I add some of the mathematical writing and reasoning is breathtaking in a certain sense, well worth doing a pure math analysis course to gain appreciation of such a book, that book being Analysis II.

[Errata added – T.]

24 October, 2018 at 10:04 am

anonymous

Chapt 1, page 16, first paragraph, bracket started at the words “Any ball”, and ends at the end of the paragraph, looking closer, just seems that there needs to be another bracket there at the end of that paragraph, 3rd ed, Analysis II electronic.I read it repeated times, there is a lot of parentheses used in that paragraph, mistake easily made.

[Corrections added – T.]

26 October, 2018 at 3:26 pm

anonymous

page 10 Exercise 1.1.8 Brackets need complete

The errors are likely to be in full stops and parentheses, possibly a few hard to place i and j symbols where there are m and n symbols, that’s my impression, I’m going to scan the entire book and write down in one total post all the observations if there are any, it’s my choice, hope it helps, cheers, best to be anonymous for this and thanks to Terry and all users for enhancing my educational experience. cheers.

I think i need to take some initiative with this, that’s four in total I am up to now in only less than reading a quarter of all the pages, so I think take the initiative, read the book in it’s entirety and that way the matter will be put to rest. I obviously had a realization and can contribute something here, best to get something done about it. kind regards.

[Correction added, thanks – T.]

26 October, 2018 at 5:38 pm

anonymous

So I checked from pages 1-71 in full, and only one further mistake was observed, seems that in paragraph one on page 64 you’ve bracketed a sentence and place the period outside the sentence, but it looks right.

Can I just say that there are more than 10,000 round and square brackets used in Analysis II, based on a search I did, so it’s actually extraordinary how little I am observing, I mean, I spelled it’s instead of its in my above post so no-one is perfect, anyway, taking a break so the quality of observing doesn’t suffer, should have the observations complete this weekend, at 70 pp in 2 hours it’s pretty easy. cheers. sorry if this isn’t very good.

[Correction added, thanks – T.]

27 October, 2018 at 5:54 pm

anonymous

ok, here is the total errata, got through the entire book, this doesn’t include any you already have listed on your list of corrections. corrected 3rd edition Analysis II …there’ quite a few, but they’re all plausible and reasonable. cheers. obviously you can delete this or do anything you see fit with it, it’s yours to keep, I gained a lot educationally doing this so it’s just appropriate to write down the observations. cheers.

PAGE10 EX 1.1.8 Brackets need complete
PAGE64 Full stop ought to be within parentheses? On first paragraph final sentence
PAGE78 Ex 4.11 Perhaps needs to finish the bracket on the exercise’s hint?
PAGE 82 (yes another one on page 82 somehow missed it) First paragraph, need to complete bracket on the second last sentence,
PAGE 83 Exercise 4.2.8 (e) period needs to be inside the parenthesis
P92 EX 4.5.1 Needs a parenthesis at the end of the hints.
P97 Defintion 4.6.12 the words “thanks to Lemma 4.6.11” have a close bracket but I couldn’t see it’s corresponding open bracket, in fact, it looks like the brackets aren’t necessarily needed for that, might just be a typo.
P102 Exercise 4.2.8 , middle sentence needs period inside parenthesis.
P103 Second paragraph second sentence actually needs a period, sentence starts with “From the quotient rule” and ends with “(why?)”.
P105 EX 4.7.9 text starting with the words “Note that the series converges” seems to have two fully bracketed sentences with respective periods outside brackets.
116 Paragraph below Remark 5.3.8 bracketed sentence beginning with “It is also possible” most likely needs the period inside the parenthesis.
125 EX5.5.3 Bracketed last sentence might be wrong hence period likely needs to be inside parentheses.
129 PARA 1, after the equation sentence beginning with the words “The notation” in brackets might need a period inside the parentheses.
135 PARA 1, the bracketed sentence ending with the “OK” perhaps needs period inside brackets?
138 Last sentence needs period inside brackets
139 PARA 2 3rd Sentence starting with “In this case“, bracketed, period needs to be inside parentheses
And in second last paragraph bracketed sentence starting with “On the other hand” same idea, period most likely needs to be inside brackets.
P146: Second paragraph bracketed sentence starting with the words “This may look a little strange”
Has period outside of an entire bracketed sentence.
P154: Last few lines you wrote “denote the function f(x) – x” but shouldn’t that function be f(x) = x?, I can’t guarantee that one because I couldn’t fully follow the lead up work but might be worth a look.
P158 PARA1, sentence beginning with the words “The portions of the circle”, bracketed, needs fullstop inside brackets.
173 EX 7.2.2 Last sentence, period outside of a full bracketed sentence
P187 EXample 8.1.2, The last sentence has a bracket at the end that appears not to have any associated open bracket.
P206 last paragraph –Bracketed sentences have period outside of brackets

[Errata added, thanks – T.]

8 January, 2019 at 4:51 am

Anonymous

Minor typo: In page 5 for the third edition (Springer) you wrote: “… where p \in [1,+\infty], but we will …” it should be “… where p \in [1,+\infty), but we will …” with a parenthesis instead.

[Actually, I do wish to allow for the possibility here that $p = \infty$ , as indicated by the usage of the $\ell^\infty$ norm. -T]

25 February, 2019 at 1:15 am

Petrus Bianchi

Dear Professor Tao,

Exercise 13.5.8 asks us to show that $\omega_1+1$ is compact but not sequentially compact. I have been able to prove the compactness part, but was unable to construct a sequence that has no convergent subsequence. So I was looking around and came across this: https://dantopology.wordpress.com/2010/07/18/sequentially-compact-spaces-i/. This link argues that $\omega_1+1$ is sequentially compact, and I haven’t been able to find any mistake in the argument so far. Am I missing something?

Thanking you in advance.

25 February, 2019 at 11:52 am

Terence Tao

I’m a little confused because Exercise 13.5.8 does not contain any reference to $\omega_1+1$ in the second edition. (I don’t have a copy of the first edition handy.)

EDIT: On the other hand, it seems that the cocountable topology described in this exercise is not, in fact, compact. As such the exercise does not serve its pedagogical purpose and should be deleted.

25 February, 2019 at 10:33 pm

Petrus Bianchi

Dear Sir,

Thank you for your reply. I think I have confused myself too. Sorry for that. Exercise 2.5.8 from the Third Edition of Analysis II is this:

Show that there exists an uncountable well-ordered set $\omega_1 +1$ that has a maximal element $\infty$, and such that the initial segments $\{x \in \omega_1 +1 : x < y\}$ are countable for all $y \in \omega_1 +1 \setminus \{\infty\}$. (Hint: Well-order the real numbers using Exercise 8.5.19, take the union of all the countable initial segment, and then adjoin a maximal element $\infty$.) If we give $\omega_1 +1$ the order topology (Exercise 2.5.5), then show that $\omega_1 +1$ is compact; however, show that not every sequence has a convergent subsequence.

The compact part is okay. But I think $\omega_1 +1$ is sequentially compact as well. If a sequence in it is considered which has infinitely many terms equal to $\infty$, then we of course have the constant subsequence $\infty, \infty, \ldots$, which converges to $infty$. If this is not the case, then we can find a subsequence $(x_n)$ (say) all of whose terms belong to $\omega_1 +1 \setminus \{\infty\}$. Now $Y=\{y_n : n\}$ is at most countable and $\omega_1 +1 \setminus \{\infty\}$ is uncountable. So we can find a minimal element of $\omega_1 +1 \setminus \{\infty\}$ that is not in $Y$. I think $(x_n)$ can be shown to converge to this minimal element. I am not exactly sure if this reasoning will work.

And one more thing. You truly are an exceptionally good teacher. Thank you.

24 March, 2019 at 1:23 am

Anonymous

Dear Prof. Tao,

don’t know if that has already been mentioned, but in the proof of Clairaut’s theorem (page 148) I think absolute value $|x| \leq 2\delta$ should be replaced with norm $\|x\| \leq 2\delta$ since $x$ is a vector.

24 March, 2019 at 9:54 am

SkysubO

Dear Professor Tao,
Analysis II，first edition，on page 418
Exercise 12.5.11 : “Let (X,d) have the property that every open cover of X has a finite subcover. Show that X is compact. (Hint: …..) ”

I guess this exercise requires the axiom of choice. However, it ‘s not mentioned in the hint.
I would be very grateful if you have time to check it !

Best regards,
Su

25 March, 2019 at 1:43 pm

Terence Tao

Generally in analysis one is willing to use the axiom of choice as needed. For this particular exercise though, it is possible to select the radii $\varepsilon$ associated to a point $x$ without recourse to choice (for instance, one could set $\varepsilon$ to be $1/m_x$ where $m_x$ is the least natural number for which $B(x,m_x)$ contains only finitely many elements of the sequence $x^{(n)}$ ).

27 March, 2019 at 9:21 am

SkysubO

Thanks for the reply ! I’ve got a general idea of the explanation yet still have a question about the existence of $m_x$ .
What if there’s no such natural number ? For example, for some point $x$ , unfortunately, the corresponding radii $\varepsilon$ one could find are all less than 1. In this situation, I can’t get $m_x$ because the corresponding set of natural numbers is empty.
How to deal with this ? Or where do I get things wrong ? Thanks !

27 March, 2019 at 11:50 am

Terence Tao

For any $\varepsilon > 0$ , one can find at least one $m$ such that $1/m \leq \varepsilon$ (the Archimedean principle).

28 March, 2019 at 7:19 am

SkysubO

Oh, I see! Thank you so much!

12 April, 2019 at 9:32 am

The Dark Knight

Dear Professor Tao,

Maybe I find a flaw in Definition 14.1.1 (Limiting value of a function): “Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, let $E$ be a subset of $X$ , and let $f:X\rightarrow Y$ be a function. If $x_0\in X$ is an adherent point of $E$ , ….“(also Definition 9.3.6 (Convergence of functions at a point) in Analysis I )

According to this definition, $x_0$ must belong to $X$ , which is the domain of $f$ . That is to say, when we talk about “the convergence of a function $f$ at $x_0$ “, the function $f$ must be defined at the point $x_0$ !

I think the condition is too strong, isn’t it ? Also, it’s not consistent with our usual perception of the notion “convergence of a function $f$ at $x_0$ “.
Professor Tao, what do you think of this? Can this condition be weakened? Thanks!

[This is a typo: $f$ should be defined on $E$ rather than $X$ . Similarly for some later statements in this section. -T]

14 April, 2019 at 8:27 am

The Dark Knight

Dear Professor Tao,

Thanks for the note!

One more question is needed for me to make this definition completely clear (please excuse me -_-).

To clarify this def, there are at least two possible ways:
1) Simply change $f:X\rightarrow Y$ to $f:E\rightarrow Y$ ;
2) Change $f:X\rightarrow Y$ to $f:X'\rightarrow Y$ , $X'$ is the domain of $f$ such that $E\subseteq X'\subseteq X$ .

Professor Tao,what’s your choice? Which one is closer to your original meaning? Many thanks!

Best regards,
D.K

20 April, 2019 at 1:12 am

Cebiş mellim

Exercise 5.5.4 requires us to work with derivative of a complex valued function, although the notion of derivative in $R^2$ is only introduced in the next chapter.

[Erratum added – T.]

20 April, 2019 at 11:30 pm

SkysubO

Dear Professor Tao,

In the first part of exercise 14.1.1, I think when $x_0$ is an isolated point of $E$ , i.e. not a limit point of $E$ , statement ” $\lim_{x\to x_0;x\in E} f(x)$ exists” isn’t logically equivalent to statement ” $\lim_{x\to x_0;x\in E\backslash\{x_0\}} f(x)$ exists and is equal to $f(x_0)$ “.

In this case, $\lim_{x\to x_0;x\in E} f(x)$ exists, but $\lim_{x\to x_0;x\in E\backslash\{x_0\}} f(x)$ doesn’t exist since $x_0$ is not an adherent point of $E\backslash\{x_0\}$ . Am I right? Thanks!

Best wishes,
SkysubO

[Correction added, thanks – T.]

4 May, 2019 at 7:57 am

cebismellim

Dear Prof. Tao,

Best regards,
Cebiş

[Corrected, thanks – T.]

5 May, 2019 at 9:30 pm

Quentin

Dear Professor Tao,

Regarding Theorem 1.5.8:

1. Page 23: do you mean $\left(V_{\alpha}\right)_{\alpha\in A}$ instead of $\left(V_{\alpha}\right)_{\alpha\in I}$ ? $I$ is used in the statement but $A$ is used in the proof. If so, you would also need to replace a finite subset $F$ of “ $I$ ” with “ $A$ “.
2. Page 24: In case 2, do you mean $B\left(y,r_{0}/2\right)\subseteq V_{\alpha}$ instead of $B\left(y,r_{0}/2\right)\in V_{\alpha}$ ?

Best,
Quentin

[Erratum added, thanks – T.]

9 May, 2019 at 2:23 am

Petrus Bianchi

Dear Professor Tao,

In Exercise 7.2.2 (of Analysis II: Third Edition) if I take $n=m=1$ and $A=\mathbb{R}, B=\mathbb{Q},$ the right hand side of the inequality $m^*_{n+m}(A \times B) \leq m^*_n(A) m^*_m(B)$ becomes $+\infty \cdot 0$ . It seems some restrictions on $A, B$ are needed. Or am I missing something?

With regards
Petrus

9 May, 2019 at 7:42 am

Terence Tao

This is fine; ${\bf R} \times {\bf Q}$ has outer measure $0 = \infty \cdot 0$ .

9 May, 2019 at 12:27 pm

Petrus Bianchi

So we can in fact define $\infty\cdot 0=0$. That’s where I had a doubt. Now it’s alright. Thanks for your reply Sir.

Regards,
Petrus

2 June, 2019 at 1:39 am

cebismellim

Dear Prof. Tao,

in Lemma 7.4.10 you are working with open sets on $\mathbb{R}^n$ but you do not specify which metric you use. However, since you mention Pythagoras’ Theorem in the proof, I assume that the theorem only holds for Euclidean and $L^p$ metrics on $\mathbb{R}^n$ ?

Regards,
Cebiş mellim

2 June, 2019 at 8:41 am

Terence Tao

All these metrics generate the same topology of open sets (cf. Proposition 1.1.18), so the precise choice of metric is not relevant for Lemma 7.4.10.

2 July, 2019 at 1:52 am

cebismellim

Dear Prof. Tao,

a very tiny typo: p.189, Lemma 8.1.9. the summation index on the right hand side is $N$ and not $n$ .

[Erratum added, thanks – T.]

14 July, 2019 at 12:17 am

Yaver Gulusoy (cebismellim)

Dear Prof. Tao,

the following is not a typo, but rather refers to the enumeration of the exercises.

Exercise 5.2.2 is followed by Excercise 5.2.4. After unsuccessfully trying to solve 5.2.2., I noticed that it seems to be impossible to do so without using the results of Exersice 5.2.4.

If this is true, and since students usually solve the exercises in the given order, wouldn’t it be better to put Exercise 5.2.2. after Exersice 5.2.4.?

[Fair enough; I added this suggestion to the errata. -T]

16 July, 2019 at 1:30 am

Yaver Gulusoy (cebismellim)

Dear Prof. Tao,

on page 197 (proof of the Proposition 8.2.10) after the third expression you justify the equality of simple integrals by mentitioning the results of Lemma 8.1.9. I think that this should rather be Proposition 8.1.10 (b) and (d).

[Correction added, thanks – T.]

16 July, 2019 at 4:05 pm

Andrew Verras

Dear Prof. Tao,

In Ex. 1.5.10, I believe in order to have the empty set be totally bounded, “there exists a positive integer” needs to be replaced by “there exists a natural number,” (when referring to the number of balls that cover X) since there are no balls in the empty set. Without doing this, part (b) is false.

[Corrected, thanks – T.]

20 August, 2019 at 8:00 pm

Issa Rice

For proposition 18.3.1:

“So any two cosets are either identical or distinct” — I think “distinct” should be “disjoint”.

“constrution” should be “construction”.

I am also a little confused with the proof, since it doesn’t seem like it shows that the sets $q + E$ are disjoint (the proof does show that distinct cosets are disjoint, and the proof for $q + E$ is similar, but it seems like we have to explicitly state this).

[Correction added, thanks -T.]

20 August, 2019 at 8:13 pm

Andrew Verras

Dear Prof. Tao,

For Proposition 2.3.2 (maximum principle), the final sentence needs $X$ to be non-empty, since a function can only achieve its maximum at $x$ if there exists some $x \in X$ .

[Correction added, thanks – T.]

29 August, 2019 at 2:25 pm

Andrew Verras

Dear Prof. Tao,

I’m sorry to continue my streak as the empty set disciple, but I’ve found another one

In Theorem 2.4.5, (c) should say X is a non-empty interval, since the empty set is not connected. Definition 9.1.1 does not exclude empty intervals.

29 August, 2019 at 8:25 pm

Andrew Verras

I suppose that still leaves the issue of (b) being vacuously true for the empty set. Perhaps instead of my first suggestion, the change should occur in the opening premise, e.g., “Let $X$ be a non-empty subset of the real line”

[Errata added, thanks – T.]

31 August, 2019 at 12:42 am

Petrus Bianchi

Dear Professor Tao

I have a doubt in the proof of Abel’s theorem. At one point we use the triangle inequality for series (Proposition 7.2.9) to conclude $|\sum_{n=0}^\infty S_{n+1}(y^{n+1}-y^n)|\leq\sum_{n=0}^\infty|S_{n+1}|(y^n-y^{n+1})$ , where $y\in[0, 1)$ . Now proposition 7.2.9 is applicable to absolutely convergent series. Although the series $\sum_{n=0}^\infty S_{n+1}(y^{n+1}-y^n)$ is conditionally convergent (we have this from the summation by parts formula and the hypothesis that $\sum_{n=0}^\infty d_ny^n$ is convergent), I don’t see how it is absolutely convergent. Thanks.

Petrus

31 August, 2019 at 1:31 am

Petrus Bianchi

Oh, please ignore the previous message. I have found the reason why that series is absolutely convergent. It’s there in the proof itself just a few lines down. Sorry about this.

Petrus

13 September, 2019 at 4:41 pm

Andrew Verras

In Definition 3.4.2, $X$ and $Y$ should be non-empty. If one of the sets is empty, then $B(X \rightarrow Y)$ is a singleton set containing only the empty function $f$ , which gives us $d_\infty (f, f) = \sup \emptyset = - \infty$ .

[Correction added, thanks -T.]

14 September, 2019 at 10:12 am

Andrew Verras

This would also be required in the premise for Proposition 3.4.4 and Theorem 3.4.5. Also possibly the final statement in Definition 3.5.5 could read “In other words, if (X, d) is a non-empty metric space,”

Also, Definition 3.4.2 states the codomain of $d_\infty$ to be the positive reals, when instead it is the non-negative reals.

An alternate fix rather than my suggestions is to define $d_\infty$ for arbitrary bounded functions $f:X \rightarrow Y$ , and to state that $d_\infty$ takes non-negative real values (and is a metric) when X and Y are non-empty.

14 September, 2019 at 10:16 am

Andrew Verras

My last paragraph may be misleading. The alternate method I mention would still require the change to the premises of Proposition 3.4.4 and Theorem 3.4.5, but it would leave Definition 3.5.5 correct as written.

26 September, 2019 at 9:41 pm

Issa Rice

For theorem 12.5.8: It seems like the proof fails to consider the case when $r_0 = +\infty$ . For example, if $X = \mathbf R$ , $Y = [0,1]$ , and $(V_\alpha)_{\alpha \in A} = (\mathbf R)$ (i.e. the collection consisting of the single open set $\mathbf R$ ), then $r(y) = +\infty$ for each $y \in Y$ so $r_0$ would also be $+\infty$ . This would go to case 2 in the proof, but case 2 computes $r_0/2$ , which is undefined.

[Correction added, thanks – T.]

2 October, 2019 at 10:39 am

Andrew Verras

In Exercise 4.2.8 (3rd Edition, p. 83), I believe that as parts (c), (d), and (e) are written, the sufficient conditions to apply Fubini’s theorem in part (e) are not met. It is of course possible that the problem is my lack of knowledge of all the theorems regarding absolute convergence in double summations.

Part (c) has us prove $d_m = \sum_{n=m}^\infty {n \choose m}(b-a)^{n-m}c_n$ is absolutely convergent for all $m \geq 0$ , part (d) has us prove for all positive $\epsilon$ less than $s$ there exists positive $C$ such that $|d_m| \leq C(s - \epsilon)^{-m}$ , which we use in part (e) to prove $\sum_{m=0}^\infty d_m(x - b)^m$ is absolutely convergent.

It seems these steps were so that we could apply Fubini’s theorem to $\sum_{m=0}^\infty d_m(x - b)^m = \sum_{(n, m)} {n \choose m}(b-a)^{n-m} c_n (x - b)^m$ . However, I think that we have not proven this series is absolutely convergent. While the series defining $d_m$ is absolutely convergent, in general we have $|d_m| \leq \sum_{n=m}^\infty {n \choose m} |b-a|^{n-m}|c_n|$ , so proving $\sum_{m=0}^\infty d_m(x - b)^m$ is absolutely convergent does not necessarily prove $\sum_{(n, m)} {n \choose m}(b-a)^{n-m} c_n (x - b)^m$ is absolutely convergent.

I fixed the apparent issue in my own notes by slightly changing the statements of parts (d) and (e). Part (d) became:

“If we define $d_m^{+} := \sum_{n=m}^\infty {n \choose m}|b-a|^{n-m}|c_n|$ , then show … $d_m^{+} \leq C(s-\epsilon)^m$ for all integers $m \geq 0$ ”

And the first sentence of part (e) became:

“Show that the power series $\sum_{m=0}^\infty d_m^{+}(x - b)^m$ is absolutely convergent.”

As far as I can tell, this seems to be the actual sufficient condition to apply Fubini’s theorem.

[Correction added, thanks – T.]

29 October, 2019 at 9:26 am

Brett Lane

In exercise 13.5.14, can we assume that the topological space is Hausdorff?
This seems necessary to prove part a.

29 October, 2019 at 11:56 am

Brett Lane

Also, in exercise 13.5.17, can we assume that $\mathbb{R}$ has the standard topology on it?

[Yes – T.]

7 November, 2019 at 6:59 pm

Pham Manh Hiep

Dear Prof. Tao,

Do you have any assumption about the metric of space $R^2$ and R in exercise 2.2.4?

7 November, 2019 at 7:06 pm

Pham Manh Hiep

Never mind. I just found you mentioned in example 1.1.4 and 1.16 about metrics of the two spaces.

Thanks,

26 November, 2019 at 7:10 pm

Pham Manh Hiep

Dear Prof. Tao,
The exercise 3.1.1 seems to assume implicitly function f is continuous at $x_0$ , because it requires to prove the function converges to f( $x_0$ ) as x converges to at $x_0$ . Could you please take a look?

[I believe the exercise is correct as stated; the hypotheses already assume a certain amount of convergence at $x_0$ that are sufficient to establish the conclusions. -T]

27 November, 2019 at 6:28 am

Brett Lane

In exercise 17.1.4, wouldn’t it be better to use the Cauchy-Schwarz inequality. Assuming that we have $||x||:=(\sum_{i=1}^{n}x_i^2)^{1/2}$ .

[This approach also works, and gives comparable results -T.]

7 December, 2019 at 8:06 am

Brett Lane

In your proof of the inverse function theorem (p. 565 of the first edition), you use the fundamental theorem of calculus on the function $\it{g(x)}$ , but this is not a real-valued function. What’s does it mean to integrate a vector-valued function?

[One integrates component by component, similarly to how derivatives of vector valued functions work as discussed after Definition 6.3.7. I’ll add a clarification to the errata. -T]

19 December, 2019 at 9:26 pm

Andrew Verras

In the last paragraph of page 152, “even when $f'$ is not invertible”, should be “even when $f$ is not invertible”.

In the last paragraph of page 153 “this argument shows that if $f(x_0)$ “, should say “this argument shows that if $f'(x_0)$ “.

[Correction added, thanks – T.]

6 February, 2020 at 8:47 am

Rafał Szlendak

In Remark 3.4.1 of Analysis II it is written that notion (b) of convergence is a special case of notion (a). Shouldn’t this be opposite?

[Corrected, thanks -T]

8 February, 2020 at 8:29 pm

Minyoung Jeong

I believe Exercise 2.4.7 should be “… Show that every non-empty path-connected set is connected.”, since empty set is path-connected yet not connected, which makes the statement in Exercise 2.4.7 false.

11 February, 2020 at 12:04 am

Dingjun Bian

Dear Professor Tao,

I think there is a typo in Example 1.3.1 starting from where it says”because the point 0…”. I believe that the statement should be using point $(0, 0)$ instead of just 0. If this is the case, then every place where we used point 0 should be changed to point $(0, 0)$ . To be precise, the statement should be: Viewed as a subset of $\mathbb{R}^2$ , it is not open, because the point $(0, 0)$ , for instance, lies in $E$ but not an interior point of $E$ . (Any ball $B_{\mathbf{R}^2,d_{l^2}}((0,0), r)$ will contain at least one point that lies outside of the x-axis, and hence outside of $E$ ).

Sincerely yours,
Dingjun

[Correction added, thanks – T.]

11 February, 2020 at 12:12 am

Dingjun Bian

Add to the previous post:

Also later in the paragraph, the statement would be: For instance, the point $(0,0)$ is now an interior point of $E$ , because the ball $B_{\mathbf{X},d_{l^2}\mid X\times X}((0,0), 1)$ is contained in $E$ .

Thank you.

12 February, 2020 at 9:39 am

Jorge Pena--Velez

Greeting Professor Tao,
I believe I have found an error in the text. In Remark 3.1.2 the second limit reads
$\lim_{x \in x_0; x \in E} f(x)$
I believe it is meant to say
$\lim_{x \rightarrow x_0; x \in E}f(x)$
I apologize if my LaTeX code is wrong but I hope the message is clear.

Best wishes,
Jorge Peña-Vélez

[Correction added, thanks – T.]

23 February, 2020 at 2:47 pm

Dingjun Bian

Dear Professor Tao,

I believe there is a typo for the quotation in Example 3.5.8, where it says “(e.g. by the ratio test, Theorem 7.5.1)”. I think it is supposed to be quoting (e.g. by the ratio test, Corollary 7.5.3), or (e.g. by the root test, Theorem 7.5.1).

Furthermore, I think that there is a small typo in the proof for Theorem 3.6.1, where almost toward the end, it says: “The above argument also shows that for every $\epsilon > 0$ there exists an $N > 0$ such that $|\int_{[a,b]} f^{(n)} - \int_{[a,b]}f| \le 2 \epsilon(b-a)$ . I think it should be $|\int_{[a,b]} f^{(n)} - \int_{[a,b]}f| \le \epsilon(b-a)$ . Although the original statement is still correct and it does not affect the final conclusion, but I think that from the above discussions in the proof, deleting 2 from the inequality would make the argument to flow a little bit smoother.

Thank you.

Sincerely yours,
Dingjun

[Corrections added, thanks – T.]

11 March, 2020 at 7:12 am

Dingjun Bian

Dear Professor Tao,

I think there is a small typo right under Example 5.2.6, where it says: “For instance, if $f(x) = \sin{(x)}$ , then $\left \| f \right \|_{\infty} = 1$ but $\left \| f \right \|_{2} = \frac{1}{\sqrt 2}$ .” I think instead of $f(x) = \sin{(x)}$ , it should be $f(x) = \sin{(2 {\pi}x)}$ . Since otherwise the function would not be $\mathbf{Z}$ -periodic. Hence, would not follow the definition of $L^2$ norm define above.

Sincerely yours,
Dingjun

[Correction added, thanks – T.]

11 March, 2020 at 7:14 am

Dingjun Bian

Sorry for posting multiple posts at once, but the system does not seem to pass my post when it is long:

I think the terminology “uniform metric” for $d_{\infty}$ should be introduced in the Definition 3.4.2 to avoid confusion, as we used it in the later chapters of the book. For example, right under Theorem 3.8.3: “Recall that …, with the uniform metric $d_{\infty}$ .”

[Suggestion implemented, thanks – T.]

11 March, 2020 at 7:14 am

Dingjun Bian

For Remark 4.1.9:

I think we need to add the word “uniformly” in the sentence “On the other hand, Theorem 4.1.6 (c) assures us that the power series will converge (uniformly) on any smaller interval $[a-r, a+r]$ .” While the original statement is correct, I think the sentence above is express the intended meaning of the statement.

[Correction added, thanks – T.]

11 March, 2020 at 7:15 am

Dingjun Bian

Right under Remark 4.3.3.

we have the Proof of Abel’s theorem. I think there are two typos in the end of the proof given in the book: “Since we can interchange limits and finite sums (Exercise 7.1.5), we thus have $\underset {n \rightarrow \infty} {\limsup} \sum ^\infty _{n=0}|S_{n+1}|(y^n-y^{n+1})\le \epsilon$ . (Should be $\underset {y \rightarrow 1:y \in [0,1]} {\limsup} \sum ^\infty _{n=0}|S_{n+1}|(y^n-y^{n+1})\le \epsilon$ ).” Similarly, there is the same typo in the last equation as well.

[Correction added, thanks – T.]

11 March, 2020 at 7:16 am

Dingjun Bian

Furthermore, in the second paragraph below Proposition 4.5.4.,
I think it is missing the word “strictly” before “increasing” in the sentence: “Since exp is increasing (Should be strictly increasing), it is injective….”

Thank you.

Sincerely yours,
Dingjun

12 March, 2020 at 4:44 pm

Dingjun Bian

Dear Professor Tao,

I find it a little hard to follow the fact that we suddenly changed from using $\exp{(z)}$ to $e^z$ when $z \in \mathbf{C}$ (From section 4.6 to section 4.7). We have not defined what it means to have a complex number as power yet in the book and it seems like we basically assumed the power series to be the definition of $e^z$ directly. I do not find it particularly satisfying as to keep it coherent, we should be able to extend Proposition 4.5.4: “For every real number $x$ , we have $\exp{(x)} = e^x$ .” to a complex number version.

One way I think might work around this could be defining $e^z = \exp{(z)} = \sum_{n=0}^\infty \frac{z^n}{n!}$ and then define $a^z = e^{\ln{(a)}z}$ . It seems to be a little bit backward though and I feel like there is a loss of coherency from previous discussions about exponential functions. Could you provide a little explanation on this? Thank you!

Sincerely yours,
Dingjun

[Correction added, thanks -T.]

14 March, 2020 at 10:55 pm

Dingjun Bian

Dear Professor Tao,

I think there is a typo right below Remark 5.2.8, where it displays the expression: $\underset{n \rightarrow \infty}{\lim} \int_{[0,1]} |f_n(x) - f(x)| dx = 0$ (It should be $\underset{n \rightarrow \infty}{\lim} (\int_{[0,1]} |f_n(x) - f(x)| dx)^{1/2} = 0$ ).

Thank you.

Sincerely yours,
Dingjun

[Correction added, thanks – T.]

28 March, 2020 at 3:42 am

Sundar

Dear Prof. Tao,
Going through Theorem 15.4.1 (Multiplication of Power series), I noticed that you first proved that the series $\sum_{n=0}^{\infty} \sum_{m=0}^{\infty} c_{m} (x-a)^{m} d_{n} (x-a)^{n}$ is absolutely convergent and then gone ahead using Fubini’s theorem to rearrange the series and prove it.
My doubt is as follows : Can I rearrange them just by using the fact that it is conditionally convergent?
My proof is as follows
$| \sum_{m=0}^{N1} c_{m} (x-a)^{m} - f(x)| \leq \epsilon$ is true for all $m \geq N1$ and
$| \sum_{n=0}^{N2} d_{n} (x-a)^{n} - g(x)| \leq \epsilon$ is true for all $n \geq N2$
Thus now $(f(x) - \epsilon)(g(x) - \epsilon) \leq \sum_{m=0}^{N} c_{m} (x-a)^{m} \sum_{n=0}^{N} d_{n} (x-a)^{n} \leq (f(x) + \epsilon)(g(x) + \epsilon)$ for all $N \geq max(N1,N2)$

Since the summation is finite, I can rearrange it to
$\sum_{i=0}^{2N} e_{i} (x-a)^{i}$ as required.
Now this sum tends to f(x)g(x) as $N \to \infty$ .

I am stuck in the part whether its acceptable to rearrange this way only taking the series to be conditionally convergent. Please tell me the mistake in my proof.

Thank you in advance.

28 March, 2020 at 3:44 am

Sundar

The missing part is $\leq (f(x) + \epsilon)(g(x) + \epsilon)$

28 March, 2020 at 11:29 am

Terence Tao

The product of $\sum_{m=0}^N c_m(x-a)^m$ and $\sum_{n=0}^N d_n(x-a)^n$ is not equal to $\sum_{i=0}^{2N} e_i(x-a)^i$ (the latter expression, when expanded, contains some terms that are not present in the former expreession).

28 March, 2020 at 8:17 pm

Sundar

Thank you for the reply.
I understand that it contains additional terms. But we can neglect them as $N \to \infty$
The product of those two finite series equals $\sum_{i=0}^{2N} e_i(x-a)^i$ .
But now $e_i = \sum_{m=0}^{i} c_{m} d_{i-m}$ if $i \leq N$
and $e_i = \sum_{m=i-N}^{i} c_{m} d_{i-m}$ if $i > N$
Now as $N \to \infty$ , I can neglect the part where $i > N$ .
Based on this, can I prove that the series converges to $f(x)g(x)$ ?

I am trying to generalize this idea for multiplication of conditionally convergent series.

29 March, 2020 at 3:32 am

Sundar

I apologize. I have realized the mistake in my reasoning. Request you to neglect this.

Thank you for your time Prof. Tao.

30 March, 2020 at 7:31 pm

Dingjun Bian

Dear Professor Tao,

I would like to make a suggestion about the use of notations in the book: I find the blend of the meaning for linear transformation composition and matrix multiplication to be a little bit unclear in many context, especially when the book did not mention explicitly how the notations are being used. To be more specific, in Lemma 6.1.16, we wrote $L_A L_B = L_{AB}$ instead of $L_A \circ L_B = L_{AB}$ . I think that without mention in beforehand, dropping the composition notation between two functions is a little bit confusing for the readers. We observe similar use of notations in Theorem 6.4.1, Several variable calculus chain rule. While we used the composition notation for the left hand side, we did not use it for the right hand side since they are linear transformations. However, there is still a sense of vagueness that may cause confusions. One way of resolving this potential issue could be mentioning the use of such notation as a footnote in Section 6.1. Thank you!

Sincerely yours,
Dingjun

[Suggestion added, thanks – T.]

31 March, 2020 at 5:13 am

Sundar

Prof. Tao,
You have asked in Exercise 15.5.6 to prove that natural logarithm function is real analytic in $(0,+\infty)$ . But natural logarithm function has radius of convergence 1. Can you please clarify if the question is right.
Thank you
Sundar

31 March, 2020 at 7:51 am

Terence Tao

The natural logarithm function $x \mapsto \log x$ has radius of convergence 1 when expanded around $1$ , but has other radii of convergence when expanded around other elements of $(0,+\infty)$ . To be real analytic it is not necessary that a single Taylor series expansion covers the entire domain of analyticity; it is only necessary that the function is analytic at each element of the domain (see Definition 4.2.1, or 15.2.1 in older editions).

5 April, 2020 at 11:47 pm

Dingjun Bian

Dear Professor Tao,

I think there is a typo in the second paragraph after the proof for Theorem 6.7.2. The book says “In such a situation it is not possible for $f^{-1}$ to exist and be differentiable at $x_0$ . (Should be $f(x_0)$ )”. Thank you.

Sincerely yours,
Dingjun

[Correction added, thanks – T.]

19 April, 2020 at 3:34 pm

Dingjun Bian

Dear Professor Tao,

I believe that there is a typo in the proof for the inverse function theorem. In the second paragraph, the book says that: “We remark that this argument shows that if $f(x_0)$ (should be $f'(x_0)$ ) is to invertible, then there is no way that an inverse $f^{-1}$ can exist and be differentiable at $f(x_0)$ .”
Thank you.

Sincerely yours,
Dingjun

[Erratum added, thanks – T.]

19 April, 2020 at 3:50 pm

Dingjun Bian

Dear Professor Tao,

Sorry again for multiple posting at once, here is another question that I have while reading and I forgot to mentioned above:

In the third paragraph after the statement of Definition 6.3.7 (Partial derivative), there is a discussion regarding to the fact that one can write directional derivatives in terms of partial derivatives. I find that the first sentence of this paragraph to be a little hard to follow: “From Lemma 6.3.5, we know that if a function is differentiable at a point $x_0$ , then all the partial derivatives $\frac{\partial f}{\partial x_j}$ exist at $x_0$ , and that $\frac{\partial f}{\partial x_j}(x_0) = f'(x_0)e_j$ .”

Question: Since the limit is only computed for $t > 0$ when evaluating the directional derivative, by Lemma 6.3.5, this would simply mean that the directional derivatives exists in both the positive and the negative direction of $v$ . However, it does not say anything about them being equal to each other, while to say that the partial derivatives $\frac{\partial f}{\partial x_j}$ exist at $x_0$ , it would seem like that we need the directional derivatives to be equal to each other in both directions for $e_j$ . In other words, I think that Lemma 6.3.5 would not suffice to imply that statement in the book in a obvious fashion.

I might be missing some obvious connections over here. Could you provide a some explanation on this? Thank you!

Sincerely yours,
Dingjun

20 April, 2020 at 9:06 am

Terence Tao

Lemma 6.3.5 applied to both $e_j$ and $-e_j$ shows that if $f$ is differentiable at $x_0$ , then $D_{e_j} f(x_0) = -D_{-e_j} f(x_0) = f'(x_0) e_j$ . This (combined with Proposition 9.5.3 from Analysis I) shows that the partial derivative $\frac{\partial f}{\partial x_j}(x_0)$ exists and is equal to $f'(x_0) e_j$ . I’ll add a sentence of clarification to the errata.

20 April, 2020 at 5:00 pm

Dingjun Bian

This is very helpful! Thank you!

20 April, 2020 at 6:50 pm

Jorge Peña-Vélez

Dear Professor Tao,
There are two small typos in your proof for proposition 7.2.6.
For the first one, right underneath equation (7.1) of your proof you wrote $\prod_{j=1} ^ {n}[a_i, b_i]$ . It should be $\prod _{i=1} ^ n [a_i, b_i]$ .
Another small typo, later in that same proof in the beginning of page 170 where you say “For all other values if x”, should be “for all other values of x”.

Best regards,
Jorge Peña-Vélez

[Corrected, thanks – T.]

21 April, 2020 at 7:36 am

Anonymous

In Exercise 3.8.2, if one replaces $1-x^2$ with $1-x^4$ , the “tail” of $\int_{-1}^1(1-x^4)^n\;dx$ is smaller than $\int_{-1}^1(1-x^2)^n\;dx$ , which means faster convergence. Does this imply a “better” polynomial approximation of the original function $f$ ?

21 April, 2020 at 11:38 am

Terence Tao

Actually it is the other way around: for $|x| \leq 1$ , $1-x^4$ is larger than $1-x^2$ , and so $c(1-x^4)^n$ will give a shorter, fatter approximation to the identity (of width comparable to $n^{-1/4}$ rather than $n^{-1/2}$ ).

21 April, 2020 at 12:15 pm

Anonymous

I’m a bit confused with your comment.

Let $Q_n(x) = c_n(1-x^4)^n$ , where $c_n$ is chosen so that $\int_{[-1,1]}Q_n(x)\;dx=1$ .

Then

$\displaystyle \frac{1}{c_n} \geq 2\int_0^{n^{-1/4}}(1-x^4)^n\; dx \geq 2\int_0^{n^{-1/4}}(1-nx^4)\;dx>n^{-1/4}$

So we have a uniform estimate on $\delta\leq|x|\leq 1$ :

$\displaystyle Q_n(x)\leq n^{1/4}(1-\delta^2)^n$

(One has the similar argument if one uses as in the Exercise, $1-x^2$ :
$Q_n(x)\leq n^{1/2}(1-\delta^2)^n$ )

One then let $P_n(x) = \int_0^1f(t)Q_n(t-x)\; dt$ to get
$\displaystyle |P_n(x)-f(x)|\leq 2M\int_{|x|>\delta}Q_n(t)\; dt +\frac{\epsilon}{2}\int_{|x|<\delta}Q_n(t)\;dt$
where $M$ is the max of $f$ and $\delta$ is chosen so that $|f(y)-f(x)|<\epsilon/2$ whenever $|x-y|<\delta$

Doesn’t the tail being smaller ( $n^{1/4}\leq n^{1/2}$ ) implies better approximation?

21 April, 2020 at 3:36 pm

Terence Tao

The bound on $Q_n$ here is $n^{1/4} (1-\delta^4)^n$ rather than $n^{1/2} (1-\delta^2)^n$ , which is worse in practice; even though $n^{1/4}$ is slightly smaller than $n^{1/2}$ , the fact that $(1-\delta^4)^n$ is larger than $(1-\delta^2)^n$ usually more than counteracts this.

21 April, 2020 at 12:19 pm

Anonymous

Are the steps in the book the original construction by Weierstrass as commented in Rudin’s real analysis book?

24 April, 2020 at 9:22 pm

Minyoung Jeong

Dear Professor Tao,
I believe there’s a minor typo in Lemma 7.4.5 on pg 178.
It says “If … and any set A …”, but I believe it should say “If … then for any set A …, we have …”.

[Corrected, thanks -T.]

28 April, 2020 at 10:43 pm

Dingjun Bian

Dear Professor Tao,

I think there is a typo in Example 7.2.12, where it says on the book: “Note that the above remarks and countable additivity (should be countable sub-additivity) imply that…”.

Moreover, I believe that in the third line of the first paragraph for the proof of Proposition 7.3.1, it should say: “For instance, $\sqrt{2} + \mathbf{Q}$ is a coset of the rationals $\mathbf{Q}$ (instead of $\mathbf{R}$ ) …”. Similarly, in the first line of the second paragraph for the proof, it should say: “We observe that every coset $A$ of the rationals $\mathbf{Q}$ (instead of $\mathbf{R}$ ) has a non-empty intersection with $[0,1]$ .”

Thank you.

Sincerely yours,
Dingjun

[Corrected, thanks – T.]

29 April, 2020 at 3:14 pm

Rafał Szlendak

Dear Sir,
For problem 6.6.8 in the errata, you have added a correction swapping $\min$ for $\max$ .
However, I do not see why is the exercise in the previous version incorrect.

One can prove that
$d(x_0, y_0) \leq \frac{\varepsilon}{1-c'}$ and
$d(x_0, y_0) \leq \frac{\varepsilon}{1-c}$ .

We obtain a better estimate for the smaller of c, c’, so if I’m not mistaken, the problem should stay as it was.

[Fair enough; the erratum has beem removed -T.]

3 May, 2020 at 1:30 am

Dingjun Bian

Dear Professor Tao,

I think there is a typo in the proof for Lemma 7.4.8 (Countable additivity). More specifically, right after the display “ $m^*(A \cap E) + m^*(A \backslash E) \leq \underset{N \geq 1}{\sup} m^*(A \cap F_N) + m^*(A \backslash E) \leq \underset{N \geq 1}{\sup} m^*(A \cap F_N) + m^*(A \backslash F_N)$ “. The book says: “But from Lemma 7.4.5 (More specifically, should be from Lemma 7.4.4 (d)) we know that $F_N$ is measurable, …”.

Moreover, I think that a “(why?)” should be inserted right after the display following the sentence: “Putting this all together we obtain…”. The reason I think a “(why?)” should be inserted is that I find this conclusion to be a little bit less obvious than it seems. More specifically, I think it is not immediately obvious that we can conclude the inequality we desire simply from $m^*(A \cap E) + m^*(A \backslash E) \leq \underset{N \geq 1}{\sup}{m^*(A \cap F_N)} + m^*(A \backslash F_N)$ and $m^*(A \cap F_N) + m^*(A \backslash F_N) = m^*(A)$ , as the right hand side of the last inequality only have the supremum of the first term, not the two terms combined.

Thank you.

Sincerely yours,
Dingjun

[Erratum added – T.]

4 May, 2020 at 11:10 pm

Dingjun Bian

Dear Professor Tao,

I believe that there is an unfinished typo correction for the statement of Corollary 2.2.3 (b). We should have “at $x_0$ ” deleted for all sentences saying “continuous at $x_0$ ” for part (b) of the corollary 2.2.3.

Thank you.

Sincerely yours,
Dingjun

[Corrected, thanks – T.]

15 May, 2020 at 12:01 am

Dingjun Bian

Dear Professor Tao,

I think there is a minor typo for the statement in Lemma 8.1.9, where it says: “Let $\Omega$ be a measurable subset of $\mathbf{R}^n$ , and let $E_1, ..., E_N$ are (should be “be”) a finite number of disjoint measurable subsets in $\Omega$ .” Moreover, there is an equal sign missing for the last display in the proof for this lemma (Lemma 8.1.9).

Furthermore, for the proof of Theorem 8.2.9 (Lebesgue monotone convergence theorem), there is a typo right before the 9th display, where it says: “From Proposition 8.2.6 (cdf) (should be (bce)) we have …”.

For the proof of Lemma 8.3.6, the second display should have the inequality to be flipped. Namely, $\int_{\Omega} f_n ^- \leq A - \frac{1}{n}$ should be $\int_{\Omega} f_n ^- \geq A - \frac{1}{n}$ .

For the statement in Proposition 8.4.1, we should add the condition saying $I \subseteq \mathbf{R}$ is bounded so that the Riemann integral is properly defined in our text. Moreover, for the last two displays in the proof for this proposition (Proposition 8.4.1), the set that we are integrating over should be $I$ instead of $\Omega$ .

Thank you!

Sincerely yours,
Dingjun

[Corrections added, thanks – T.]

15 May, 2020 at 7:40 pm

Dingjun Bian

Dear Professor Tao,

I think there is a typo in the third paragraph counting from the start of section 8.5 Fubini’s theorem, where it says: “Secondly, we can fix $x$ and compute a one-dimensional Lebesgue integral in $y$ , and then take that quantity and integrate in $x$ , thus obtaining $\int_{\mathbf{R}}(\int_{\mathbf{R}} f(x, y) dy) dx$ . Secondly (should be “Thirdly”), we could…”

Furthermore, I believe that there is a typo in the proof for the Fubini’s theorem (Theorem 8.5.1). To be more specific, in the fifth paragraph for the proof, it says “since one can use Lemma 8.1.4 (should be Lemma 8.1.5) to write…”

Thank you!

Sincerely yours,
Dingjun

[Erratum added, thanks -T.]

7 August, 2020 at 9:41 pm

Z Liu

Dear Professor Tao,

I think in exercise 3.1.3 in pg 48, the equivalence of prop 3.1.5(c) and (d) in topological spaces, the statement ” Furthermore, if x0 is in E, then f(x0)=L” does not hold for arbitrary topological spaces (For example, if we let Y be trivial topology on R, E=R, with standard topology, then L can be any real number).

Thanks,

[Erratum added, thanks – T.]

4 September, 2020 at 3:39 pm

Andrew Verras

Dear Prof. Tao,

I think the hint in Exercise 8.2.10 on page 200 may be slightly wrong in a way that has led to a few headaches for me. It says:

To prove it, first show that for any positive integer $m$ , we can find an $N > 0$ such that $m(\{x \in [0,1] : f_n(x) > 1/m\}) \leq \epsilon / 2^m$ for all $n \geq N$ .

I found this to be insufficient to prove the desired uniform convergence, and I had to use a slightly different approach. I think it would lead to less confusion in the future if the hint was edited to read:

To prove it, first show that for any positive integer $m$ , we can find an $N > 0$ such that $m(\bigcup_{n = N}^{m}\{x \in [0,1] : f_n(x) > 1/m\}) \leq \epsilon / 2^m$ .

[Correction added, thanks – T.]

5 September, 2020 at 6:15 am

Andrew Verras

Although I just noticed a gap in my proof is harder to fill than I thought, so at this point I don’t actually know how to prove my suggested hint. But I suspect it’s true, and it’s clear that it implies the uniform convergence if true.

5 September, 2020 at 6:32 am

Andrew Verras

I figured out how to prove my hint. It follows nicely from Exercise 7.2.3.

8 September, 2020 at 11:34 am

SkysubO

Dear Professor Tao,

As to the connectedness of the empty set, I find two contradictory defintions in Analysis I and II.

In Analysis I, Examples 11.1.3 says “The empty set, as well as singleton sets… , are connected, .. ” In Analysis II, Definition 2.4.1 says “We declare the empty set Ø as being special – it is neither connected nor disconnected; …” Why is this?

Thanks!

8 September, 2020 at 5:03 pm

Terence Tao

Good point. Definition 11.1.1 should be modified to exclude the empty set from being connected (and then Examples 11.1.3 should be modified accordingly.) One reason for making this exclusion is to ensure that an elementary set can be uniquely decomposed into connected components (up to permutation), much as the exclusion of 1 from the prime numbers ensures that every natural number can be uniquely decomposed into primes (up to permutation).

9 September, 2020 at 6:50 am

SkysubO

Thanks Professor! That explanation is very helpful!

8 October, 2020 at 4:20 pm

Anonymous

Dear Professor Tao,

In Exercise 6.6.1 of Volume 2, should $f:[a,b] \rightarrow R$ perhaps be replaced by $f:[a,b] \rightarrow [a,b]$ ?

Else, we could define $f:[1,2] \rightarrow R$ as f(x) = 5 +0.5x and f would be differentiable with $|f'(x)| \leq 1$ for every x in [1,2], but f wouldn’t actually be a contraction

[Correction added, thanks – T.]

8 October, 2020 at 4:35 pm

Anonymous

In reference to the same exercise 6.6.1, should the condition for strict contraction be 0<|f'(x)|<1, else we could have f(x) = a, and |f(x)-f(y)| = 0, which would imply that f(x) isn't a strict contraction because c (the contraction constant) needs to be greater than 0.

[This is not necessary. One can always increase the contraction constant slightly if desired. -T]

10 October, 2020 at 7:45 pm

Anonymous

Hello Professor Tao,

Does Exercise 1.4.8 require the axiom of choice?

[No – T.]

12 October, 2020 at 1:05 am

Anonymous

In my proof of part (c), I started out with a sequence $(a_n)_{n\geq 1}$ in $\overline{X}$ and then chose for each $a_n$ a representative Cauchy sequence $(x_k)_{k\geq 1}$ in $X$ such that $a_n$ is the formal limit of $(x_k)_{k\geq 1}$ . For this I used the axiom of choice.

How can this be avoided?

I tried to do some research and on the ProofWiki page https://proofwiki.org/wiki/Completion_Theorem_(Metric_Space) it also uses AC, though in a different manner.

12 October, 2020 at 10:16 am

Terence Tao

Ah, fair point: the construction of $\overline{X}$ does not require choice, but the verification of completeness does require (countable) choice. I’ll update the erratum to point this out.

13 October, 2020 at 3:32 pm

Anonymous

Hello Professor Tao,

The hint for Exercise 1.5.3 mentions Theorem 9.1.24, but I couldn’t find a way to directly use Theorem 9.1.24. Instead, to show that a subset E is closed+bounded=>E is compact, I mimicked the proof of the corresponding half of Theorem 9.1.24 by using (an appropriately generalized to R^n) Theorem 6.6.8 (Bolzano-Weierstrass): E bounded implies convergent subsequence, then E closed implies the limit of that subsequence is in E.

Perhaps the hint for Exercise 1.5.3 should mention Theorem 6.6.8 instead?

[Both approaches will work. To prove that a sequence in a closed and bounded subset of (say) ${\bf R}^2$ has a convergent subsequence, first use Theorem 9.1.24 and Proposition 1.1.18 to extract a subsequence where the first coordinates converge, then use these theorems again to extract a further subsequence where the second coordinates converge as well. -T]

13 October, 2020 at 7:00 pm

Anonymous

In Theorem 1.5.10(a), does “Z is closed” mean closed with respect to X, or with respect to Y?

[The former, though in this particular case the two properties are equivalent. -T]

16 October, 2020 at 2:59 pm

Anonymous

In Exercise 1.1.11, R^n should be an arbitrary set X.

[Correction added, thanks – T.]

22 October, 2020 at 7:08 pm

Winston Tsai

In Exercise 2.5.4, I think the trivial topology being not Hausdorff requires X to have more than one element.

[Erratum added, thanks – T.]

23 October, 2020 at 2:09 am

Winston Tsai

I think there is a typo in the fifth line of Exercise 2.5.6.

[Correction added, thanks – T.]

23 October, 2020 at 12:20 pm

Anonymous

Also in Exercise 2.5.6, if the goal is to show that the cofinite topology cannot be obtained from a metric, could one simply use the fact that metric spaces are Hausdorff?

[That would work – T.]

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