Analysis II

Last updated September 27, 2021

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 prior to the corrected third edition may be found here.

— 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 15: In Proposition 1.2.15, $\supset$ should be $\supseteq$ (two occurrences).
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 21: In Exercie 1.4.7 (b), ${\bf R}^+$ should be $[0,+\infty)$ .
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). Also, add the requirement that $r$ be finite.
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 38: In Exercise 2.4.7, “replace “every path-connected set” by “every non-empty path-connected set”. In Exercise 2.4.6, add the hypothesis that $I$ is non-empty.
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 52: In the last paragraph of the section, $f|_Y$ should be $f|_E$ .
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 “uniform metric” next to “sup norm metric” and $L^\infty$ metric”, and restrict the definition of $d_\infty$ to the case when $X$ is non-empty, then add “When $X$ is empty, we instead define $d_\infty(f,g)=0$ “; similarly for Definition 3.5.5. In Remark 3.4.1, “(b) is a special case of (a)” should be “(a) is a special case of (b)”. Finally, in Definition 3.4.2, use $[0,+\infty)$ in place of ${\bf R}^+$ .
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”. Also “ $f^{(n)}$ converges uniformly” should be “ $\sum_{n=1}^\infty f^{(n)}$ converges uniformly”.
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 76: In the first display of Example 4.1.5, $(-2^n)$ should be $(-2)^n$ .
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 79: In Definition 4.2.4, add “with the property that every element of $E$ is a limit point of $E$ ” at the end of the first sentence. At the end of the second sentence, add “, in particular $f': E \to {\bf R}$ is also a function on $E$ .”
Page 81: In Corollary 4.2.12, “ecah” should be “each”.
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 $|c_n|$ . At the beginning of the exercise, “anaytic in $a$ ” should be “analytic at $a$ “.
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 134: In Lemma 6.2.1, “ $x_0 \in E$ , and $L \in {\bf R}$ ” should be “ $L \in {\bf R}$ , and let $x_0$ be a limit point of $E$ “. In the previous display, $E \backslash \{x_0\}$ should be $E - \{x_0\}$ .
Page 135: In the first paragraph, the period should be inside the parenthesis. In Definition 6.2.2, $x_0$ should be a limit point of $E$ .
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 $n$ rather than from $1$ to $m$ . “Because each partial derivative … is continuous on $F$ ” should be “Because each partial derivative … exists on $F$ and is continuous at $x_0$ “.
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 163: after “if $A$ and $B$ are disjoint”, add “, and more generally, that $m(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty m(A_n)$ when $A_1,A_2,\dots$ are disjoint”.
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. “Since everything is positive” should be “Since everything is non-negative”, and in the preceding sentence, add “; for instance, in this chapter we adopt the convention that an infinite sum $\sum_{j \in J} a_j$ of non-negative quantities $a_j$ is equal to $+\infty$ if the sum is not absolutely convergent.”
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”. In Example 8.1.7, “the integral” should be “the interval”.
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 199: Exercise 8.2.4 should be moved to Section 8.3 (as it uses the absolutely convergent integral).
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, Kyle Hambrook, 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.

386 comments

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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.]

3 November, 2020 at 12:08 am

Anonymous

In the paragraph after Example 3.2.9, I think f|_Y should be f|_E.

[Erratum added -T.]

3 November, 2020 at 3:34 pm

Anonymous

In Definition 3.4.2, if X is empty then I think the metric outputs $-\infty$ .

[Good point; a special convention is needed in this case. -T]

4 November, 2020 at 2:40 pm

Anonymous

Similarly for Definition 3.5.5 and Example 3.5.6.

[Erratum updated – T.]

5 November, 2020 at 4:34 pm

Winston

In Definition 4.2.4 where you define k-times differentiability, it is implicitly assumed that F^(k) has domain E. Doesn’t this require every point in E to be a limit point?

This also leads me to another point. I think the “derivative as a function” has never been defined (in either Analysis I or II). Only the value of the derivative of a function at a point has been defined. That is, let F be a function from E to R. Then the function F’ has not been defined, only F'(x_0) is defined for limit points x_0 of E for which this value exists.

Given your definition of differentiability on a domain (Definition 10.1.11), I’ve been working with the ad-hoc definition that F’ is just the function defined on the set of limit points of E for which the derivative exists.

Also, I still feel that “F is differentiable on S” should mean that the derivative exists at every point of S (and hence every point of S is a limit point of E), so we can write F'(x) for any x in S. Then when one says “F is differentiable” it means that F is differentiable on E, the domain of F.
See my comment on March 13, 2020 on the Analysis I page.

[Good point; I have now added the restriction that every point of $E$ is a limit point to the definition. -T.]

5 November, 2020 at 8:42 pm

Anonymous

I apologize if this seems amateurish, I’m just trying to organize my thoughts.

Let $f: X\subseteq\mathbb{R} \to \mathbb{R}$ be a function.

We define the derivative function $f': x \in A \to f'(x)\in \mathbb{R}$ , where $A\subseteq X$ is set of limit points of $X$ for which the derivative exists.

We say that $f$ is differentiable on $S \subseteq X$ iff $f'(x)$ exists for all $x \in S$ (so $S\subseteq A$ ).
The meaning of the original Definition 10.1.11 can always be recovered by letting $S\subseteq X$ be the set of limit points of $X$ .
If $f$ is differentiable on $X$ then we just say that $f$ is differentiable.

With these modifications I think Definition 4.2.4 can remain unchanged, although one could extend the definition to “k-times differentiable on $S \subseteq E$ ” by using the phrase “on $S$ ” in the appropriate spots. The definition of “kth derivative of $f$ ” remains unchanged (this is where we need a definition of a derivative function), but when $f$ is k-times differentiable on $S$ then we could define the “k-th derivative of $f$ on $S$ ” to be $f^{(k)}|_S$ . This notion of k-times differentiable on a subset is used in Proposition 4.2.6.

Then also I think my two previous errata for Page 253 and Page 297 of Analysis I would also be reverted.

Would these definitions be worth making/modifying?

6 November, 2020 at 1:38 am

Anonymous

Actually, defining “k-th derivative of f on S” is pointless. To say that f is k-times differentiable on S is to say that S is a subset of the domain of f^(k). The function f^(k) will always exist, it’s just that it will often be the empty function.

14 November, 2020 at 12:31 am

Winston

In Example 4.1.5, for the first fraction, I think the ‘n’ should be outside the parentheses.

[Correction added -T]

17 November, 2020 at 1:15 am

Winston

In Corollary 4.2.12, ‘each’ is mispelled.

[Correction added – T.]

30 December, 2020 at 9:05 am

Anonymous

The irony…

11 December, 2020 at 6:15 pm

Anonymous

In Example 3.5.8, I think it should be that the series \sum f_n converges uniformly, not that f_n converges uniformly.

[Erratum added, thanks – T.]

11 December, 2020 at 8:25 pm

Anonymous

In Definition 3.8.6(c) is the absolute value sign on f(x) redundant?

[Technically yes; however sometimes one can consider more general approximations to the identity in which the non-negativity hypothesis in (a) is omitted, in which case it is useful to retain the absolute value sign in (c). -T]

13 December, 2020 at 11:00 pm

Anonymous

In Exercise 4.2.8, “real analytic in a” should be “real analytic at a”

[Erratum added – T.]

19 December, 2020 at 1:27 pm

Kyle Hambrook

On page 163, you give a set of reasonable properties for a measure defined on all subsets of R^n to obey: measure of unit cube is 1; finitely additive; monotone; translation invariant. The you write “Remarkably, it turns out that such a measure does not exist; one cannot assign a non-negative number to every subset of R^n which has the above properties.” This is not true. It is true if n >= 3 or if “finitely additive” is replaced by “countably additive.” But, if n=1 or n=2, there is a finitely-additive translation invariant extension of Lebesgue measure defined on all subsets of R^n. Original Reference: S. Banach, Sur le probleme de la mesure, Fund. Math. 4 (1923), 7-33.

[Erratum added, thanks – T.]

20 December, 2020 at 1:26 pm

Kyle Hambrook

Not to be too picky, but my last name is “Hambrook”, not “Handbrook” as you wrote in the “thanks for corrections” section of this post.

[Corrected, thanks – T.]

22 December, 2020 at 12:05 am

Anonymous

In Definition 6.2.2, the definition of derivative, should x_0 be a limit point of E?

[Erratum added – T.]

24 December, 2020 at 8:07 pm

Anonymous

In Analysis 1 we only defined summation on countable sets J if there is a bijection from N to J which makes the corresponding infinite series absolutely convergent. In the definition of outer measure (Definition 7.2.4) and the subsequent material, should we take the sum of volumes to be +infinity if the sum is not absolutely convergent?

[Yes; an erratum has now been added to this effect. -T]

28 December, 2020 at 7:18 pm

Anonymous

In Example 8.1.7, ‘integral’ should be ‘interval’.

[Corrected added, thanks – T.]

29 December, 2020 at 1:54 am

Anonymous

In Exercise 8.2.4, I think we have not yet defined the integral for functions with negative values. Should we use Definition 8.3.2?

[Good point; erratum added. -T]

29 December, 2020 at 4:56 pm

Anonymous

In Proposition 8.3.3, should f and g take values in the extended reals?

31 December, 2020 at 11:59 am

Terence Tao

In principle yes, but a technical difficulty arises in Proposition 8.3.3(b) in this case because $f+g$ might not always be everywhere defined. One can fix this by working with functions that are only defined almost everywhere rather than everywhere (and extending the Lebesgue integral to such absolutely integrable almost everywhere defined functions, taking advantage of Proposition 8.3.3(d)), and changing all the hypotheses in Proposition 8.3.3 to only be applied almost everywhere. This is done in my book on measure theory but for this text I am only giving the briefest introduction to the Lebesgue integral and chose not to dwell on these subtleties.

5 February, 2021 at 9:31 am

Anonymous

In Proposition 3.3.3, can one drop the assumption of completeness of $Y$ ? I can’t find a counterexample.

~~[Yes, you are right; I’ve added an erratum to this effect. -T]~~

5 February, 2021 at 2:51 pm

Anonymous

One can show that this is a Cauchy sequence using the assumptions that for each $n$ , $\lim_{x \rightarrow x_{0} ; x \in E} f^{(n)}(x)$ exists and $f^{(n)}$ converges to $f(x)$ uniformly.

Without completeness of $Y$ , how can one show that the limit of the sequence $\left(\lim_{x \rightarrow x_{0} ; x \in E} f^{(n)}(x)\right)_{n=1}^{\infty}$ exists?

If one drops the completeness of $Y$ , does one needs to assume that $\lim _{x \rightarrow x_{0} ; x \in E} f(x)$ exists (instead of a conclusion)?

5 February, 2021 at 4:14 pm

Terence Tao

Ah, that is correct, one runs into issues without completeness after all. A counterexample is $X = [0,1]$ , $E = (0,1)$ , $Y = (0,2)$ , $x_0 = 0$ , $f_n(x) := x+1/n$ , $f(x) := x$ .

18 May, 2021 at 5:47 pm

sam

Dear Professor Tao:
For Exercise 1.4.8(c): Given a Cauchy sequence of formal limits in the complete metric space, I was trying to show that it converges to the sequence of diagonal elements, but it did not seem to work. Does one have to use the axiom of choice for this part ? I’d be appreciated if you can provide some further hints.

[Yes; see errata for the third edition. -T]

26 May, 2021 at 10:21 am

Dear Professor Tao,

in the third ed., p. 14, Prop. 1.2.15 g

for the intersection $\cap_{\alpha\in I}{F_\alpha}$ the index set $I$ should be assumed to be non-empty.

[This is not necessary, as by definition the intersection is $X$ when $I$ is empty. -T]

7 June, 2021 at 1:57 pm

Thanks for the reply.
I know, that one can define the intersection as $X$ when $I$ is empty.

But in Analysis I, third ed., p. 60 you defined the intersection only for non-empty index sets:

“Note that if $I$ was empty, then
$\bigcup_{\alpha\in I}{A_\alpha}$ would automatically also be empty (why?).
We can similarly form intersections of families of sets, as long as the
index set is non-empty. More specifically, given any non-empty set $I$,
and given an assignment of a set $A_\alpha$ to each $\alpha\in I$, we can define the intersection $\bigcap_{\alpha\in I}{A_\alpha}$ by first choosing some element $\beta$ of $I$ (which we
can do since $I$ is non-empty), and setting
$\bigcap_{\alpha\in I}:=\{x\in A_\beta: x\in A_\alpha\forall\alpha\in I\}$,
which is a set by the axiom of specification.”

So I would recommend to add a remark somewhere, how the intersection should be defined in the case of $I=\emptyset$.

One can find a good explanation here:
https://en.wikipedia.org/wiki/Intersection_(set_theory)#Nullary_intersection

27 May, 2021 at 3:09 am

Dear Professor Tao,

in the third ed., p. 21, Ex. 1.4.8. (b)

you define a metric with codomain $\mathbb{R}^+$.

In Analysis 1, Ex. 5.5.3, you defined this set as the set of all positive real numbers.

Shoudn’t the codomain include 0?

Same problem: Analysis 2, Def. 3.4.2

[Errata added, thanks – T.]

28 May, 2021 at 5:31 pm

sam

Dear professor Tao:
Thank you for your reply to my question above on the Axiom of Choice and Exercise 1.4.8(c). However, I wonder if one can circumvent the use of Choice as follows: Given a Cauchy sequence in $\overline{X}$ , the $k^{th}$ term corresponds to a Cauchy sequence $(\bar{x}_{ki})_{i}$ in X, which permits some integer after which the terms are of distance at most $1/k$ apart, take the minimum of such integers and denote it by $n_{k}$ (Assuming $n_{k}$ to be increasing). We then consider the diagonal sequence formed by picking the $n_{k}^{th}$ term of the $k^{th}$ member of the original sequence in $\overline{X}$ . May I know what is the issue with this argument?

7 June, 2021 at 12:57 pm

Terence Tao

To make this argument rigorous one would have to select a representative Cauchy sequence $(\overline{x}_{ki})_i$ attached to each formal limit $\overline{x}_k = \mathrm{LIM} (\overline{x}_{ki})_i$ , and this is where the axiom of (countable) choice needs to be invoked.

9 June, 2021 at 9:39 am

Dear Professor Tao,

in Ex. 2.4.7 one is asked to prove that every path-connected subset E of X is connected.

The empty set is path-connected (vacuous truth), but in Def. 2.4.1 you declare the empty set as being special (neither connected nor disconnected.

[Erratum added to remove the empty case – T.]

17 June, 2021 at 6:50 am

William Deng

Dear Prof. Tao
I’m wondering about the footnote on the page containing Proposition $1.1.18$ where you talk about the “automobile” and “pedestrian” metrics. While I understand that given two points $\textbf{x}$ and $\textbf{y}$ in the city (modelled as a subset of $\textbf{R}^2$ ), the automobile distance between $\textbf{x}$ and $\textbf{y}$ is generally less than the pedestrian distance between $\textbf{x}$ and $\textbf{y}$ , for the purposes of discussing convergence of sequences with respect to different metric spaces (which is what the footnote refers to), I think any sequence $(\textbf{r}^{(n)})_{n=1}^{\infty}$ of points converging in the automobile metric (so that the driving time goes to zero) will also converge in the pedestrian metric (so that the walking time goes to zero), and vice versa (it’s just that the “rate” of the convergence might be different).

19 June, 2021 at 10:18 am

Terence Tao

The example here was explicitly “whimsical” in nature and so should be interpreted loosely. For instance, imagine a person walking towards another person separated by some low fence or other barrier, until they come in contact. Informally this is representing convergence in the pedestrian metric, but if one used cars instead of people then one would not have convergence in the automobile metric if the automobile cannot (legally) cross the fence.

22 June, 2021 at 2:09 pm

Anonymous

In Stein-Shakarchi’s Fourier Analysis, The series $\sum_{n=-\infty}^{n=-1}\frac{e^{in\theta}}{n}$ is given as an example that is not the Fourier series of a Riemann integrable function. They argue by assuming it is the Fourier series for $f$ and the Abel mean at $\theta=0$ is
$\displaystyle |A_r(f)(0)|=\sum_{n=1}^\infty\frac{r^n}{n}\quad (0\le r< 1)\quad (1)$
They write that (1) tends to infinity as $r$ tends to $1$ because $\sum 1/n$ diverges and this contradicts $|A_r(f)(0)|\le \sup_{\theta\in[-\pi,\pi]} |f(\theta)|$ .

1. But what they are saying is $\sum_{n=1}^\infty\lim_{r\to1}\frac{r^n}{n}$ , not $\lim_{r\to1}\sum_{n=1}^\infty\frac{r^n}{n}$ . How can one justify exchange of the limit and the sum?

2. If one relaxes the Riemann integrable condition to Lebesgue integrable, can one find such $f$ ?

3. Given a sequence $(a_n)$ , are there well-known results for telling whether it is the Fourier coefficients of some function $f$ ?

23 June, 2021 at 2:14 am

ugroh

Regarding your question 3.: There is a paper by Carathéodory from 1918. It is written in german (of course) and you find it in Mathematische Zeitschrift, 1918 on 309 – 320. There is a reference (footnote 1) answering your question (I hope).

23 June, 2021 at 10:07 am

Anonymous

For question 1, since $r=e^{i n\theta}$ , thus $|r|\leq 1$ , so $\frac{r^n}{n}$ is dominated by $1/n$ , apply the dominated convergence theorem, we can interchange limit and summation.

23 June, 2021 at 10:25 am

Anonymous

Actually, never mind. The explanation above is not valid. I don’t think we are exchanging limit and sum here. We just need to find a value of $r$ such that the summation diverges.

26 June, 2021 at 6:01 am

Dear Professor Tao,

in the third ed., p. 77 (Theorem 4.1.6. (d)) you write:

… converges uniformly to f ‘ on the interval [a-r,a+r].

It should rather be:

… converges uniformly to f ‘(x) on the interval [a-r,a+r].

26 June, 2021 at 6:09 am

Dear Professor Tao,

in the third ed., p. 82 – Ex. 4.2.6 – one is asked to show that every polynomial P(x) of one variable is real analytic on the real numbers.

In Def. 4.2.1 you define real analytic functions.

A polynomial is an algebraic object, but not a function.

It only induces a function, as you can define a polynomial function by evaluating the polynomial.

So it should rather be:

Using Ex. 4.2.5, show that every polynomial function of one variable is real analytic on R.

[In these texts we often “abuse notation” by using a formal expression, such as a polynomial, to refer to the function it induces; see e.g., the discussion after Example 3.3.3 in Analysis I. -T]

15 July, 2021 at 10:21 am

William Deng

Dear Prof. Tao, shortly after stating Theorem $1.5.7$ , you say that “However, a version of the Heine-Borel theorem is available if one is willing to replace closedness with the stronger notion of completeness…”, but actually, I think intrinsic closedness is equally strong as completeness (i.e. from Proposition $1.4.12(a)$ , we see that every complete metric space is also intrinsically closed, while combining $1.4.12(b)$ with Exercise $1.4.8$ reveals that every intrinsically closed metric space is also complete).

[Completeness is indeed equivalent to being intrinsically closed, but this is not what being closed refers to by default. -T]

15 July, 2021 at 6:25 pm

William Deng

A suggestion: for clarity, in Definition $1.2.1$ , replace “ $r>0$ ” by “ $r\in{(0,+\infty)}$ ” or “ $r\in\textbf R^+$ ” or “finite $r>0$ ” or something to that effect (basically, you probably don’t intend $r=+\infty$ to be a legitimate value for $r$ , otherwise for instance one can show using Definition $1.5.3$ that every metric space is bounded which would make the concept rather boring).

[The requirement that $r$ be finite will be added to Definition 1.5.3. -T.]

16 July, 2021 at 7:07 am

William Deng

Is it possible to weaken the hypotheses in Corollary $1.5.9$ as follows: only assume $K_1$ to be compact and weaken the assumption of compactness on the other sets $K_2,K_3,...$ to closedness (in $K_1$ )? This is a strictly weaker assumption because although one can deduce from the compactness, hence boundedness of $K_1$ along with the totally ordered nature of the sets $K_1\supseteq K_2\supseteq K_3\supseteq ...$ that the other sets $K_2,K_3,...$ are also closed (in $K_1$ ) and bounded, we saw previously that the full Heine-Borel theorem is not necessarily true for an arbitrary metric space.

16 July, 2021 at 8:12 am

William Deng

Actually never mind, that would not be strictly weaker, rather it would just be an equivalent formulation, thanks to Theorem $1.5.10.$ (so essentially, it seems that the full Heine-Borel theorem can be recovered when working inside a compact metric space).

17 July, 2021 at 1:04 pm

Simon

I believe one of the corrections has a small typo. The correction of the beginning of the proof of Theorem 6.3.8 on page 140 has $n$ and $m$ swapped, i.e. it should say “[…] the sum on the RHS should be from $1$ to $n$ rather than from $1$ to $m$.”

[Corrected, thanks – T.]

22 July, 2021 at 12:37 am

Kevin

Dear prof. Tao, I believe the statement “Furthermore, if $x_0 \in E$ , then $f(x_0) = L$ ” in Proposition 3.1.5 is wrong.
Consider the function $f : [0, 1] \to {\bf R}$ where $f(0.5) = 100$ and $f(x) = x$ for every $x \neq 0.5$ .
Even if we can construct the function $g$ , we cannot derived the continuous property for $f$ at $x_0$ .

[For this example all four parts (a), (b), (c), (d) do not hold, precisely because of this “Furthermore, …” condition in (d). If one omitted this extra assertion then one would no longer get the logical equivalence of the four statements. -T]

5 September, 2021 at 5:25 am

William Deng

Exercise $2.4.6$ is technically false when the set $X$ is non-empty while the index set $I$ is empty. Perhaps the additional assumption that $I$ be non-empty should be inserted?

[Erratum added, thanks – T.]

13 September, 2021 at 12:37 am

Kevin

Dear prof. Tao, thank you for your reply of my incorrect statement above.
I have another question, and it’s about exercise 5.2.6(d).
Since we are restrict our functions $f_n$ and $f$ to 1-periodic, we cannot have arbitrary large integral on $[0,1]$ when $f_n$ is also pointwise converges to $f$ (this statement is similar to exercise 5.2.6(a), but I have not come up with a proof yet).

[Note in part (d) that the functions $f_n$ are not required to be uniformly bounded, in contrast to (a). -T]

15 September, 2021 at 7:34 pm

Kevin

Dear prof.Tao, Definition 5.4.5 seems to be ill-defined, since $f(x)$ is a complex number, thus we cannot compare $f(x)$ with $0$ and $\varepsilon$ .
Also, in Definition 3.8.6, one already given $f(x) \geq 0$ , is the absolute value in condition (c) still necessary?

21 September, 2021 at 12:12 am

Kevin

Dear prof. Tao, I think in Theorem 6.3.8, page 140, the statement “Because each partial derivative $\frac{\partial f}{\partial x_j}$ is continuous on $F$ ” should be “Because each partial derivative $\frac{\partial f}{\partial x_j}$ exists on $F$ and is continuous at $x_0$ ” .

[Erratum added, thanks – T.]

12 October, 2021 at 11:53 pm

sin

Dear Professor Tao:
Can you provide some further hints on how to prove Theorem 4.1.6(d) using Theorem 3.7.1?

[Worl with the partial power series $\sum_{n=0}^N c_n (x-a)^n$ and their derivatives. -T]

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