Last updated Jan 27, 2022

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 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 91: Before Definition 4.5.5, “exp is increasing” should be “exp is strictly increasing”.
• 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).  In Example 7.2.9, “each rational number ${\bf Q}$” should be “each rational number $q$“.
• 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 ${\bf Q}$“.  In Exercise 7.2.5, “Q1” should be “Exercise 7.2.3”.
• 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 176: In the proof of Proposition 7.3.3, “cardinality 3n” should be “cardinality $3n$“.
• 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 181: “… on our wish list is (a)” should be “… on our wish list is (i)”.
• 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.

— Errata for the fourth edition —

• Page 4: In Example 1.1.9, $\sup(3,4)$ should be $\sup\{3,4\}$.
• Page 14: In Proposition 1.2.15(c), delete the word “then”.
• Page 42: In Exercise 2.5.4, when referring to the trivial topology being non-Hausdorff, add the hypothesis that the space contains at least two points.
• Page 53: In Exercise 3.2.2(c), Lemma 7.3.3 should be Lemma 7.3.3 from Analysis I.
• Page 55: In Remark 3.3.7, “it only works” should be “they only work”.
• Page 107: “Napoleons” should be “Napoleon’s”.
• Page 118?: in the proof of Lemma 5.4.6, the reference to Lemma 7.3.3 should instead be Exercise 7.3.2 from Analysis I.
• Page 150: In exercise 6.6.7, Lemma 7.3.3 should be Lemma 7.3.3 from Analysis I.
• Page 167: “for every box $j$” should be “for every box $B_j$“.
• Page 170: In Example 7.2.11, “$m^*({\bf R})$ has outer measure” should be “${\bf R}$ has outer measure”.
• Page 182: In Corollary 7.5.7, “then so is” should be “then so are”.
• Page 187: In Example 8.1.7, “simple integral” should be “Lebesgue integral”.
• Page 205: In Theorem 8.5.1, “Then there exists” should be “Then there exist”.

Caution: the page numbering is not consistent across editions.  Starting 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, William Deng, 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, Frédéric Santos, 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.