《陶哲轩实分析》部分勘误

我在读《陶哲轩实分析》，作者是陶哲轩,译者王昆扬.2008年11月第一版，第一次印刷.我在此添加一部分中译本印刷错误，若网友发现了另外的错误，请在评论里补充，由我代为添加.若有不当之处，敬请指正.

勘误：

3.2节，37页,公理3.9中，“不同”应该改为“不交”.

3.4节，48页，习题3.4.10中，$(\bigcap_{\alpha\in I}A_{\alpha})\bigcap(\bigcap_{\alpha\in J}A_{\alpha})=(\bigcap_{\alpha\in I\bigcap J}A_{\alpha})$应当改为$(\bigcap_{\alpha\in I}A_{\alpha})\bigcap(\bigcap_{\alpha\in J}A_{\alpha})=(\bigcap_{\alpha\in I\bigcup J}A_{\alpha})$

7.1节，130页，命题7.1.11，(e),$f:X\bigcap Y\to \mathbb{R}$应该改为$f:X\bigcup Y\to\mathbb{R}$.

7.4节，145页，习题7.4.1，$\sum_{m=0}^Ma_{f(m)}$应当改为$\sum_{m=0}^{\infty}a_{f(m)}$.

7.5节，145页，(b):如果$\alpha>1$,那么级数$\sum_{n=m}^{\infty}a_n$是条件发散的（从而是绝对发散的）。我认为最好把（从而是绝对发散的）改为（从而不是绝对收敛的），后者似乎更符合原文.

8.5节，171页，习题8.5.15中，“A的基数不小于或等于B的基数”这句话有歧义.到底是“不小于”或等于呢，还是不“小于或等于”呢？结合题目意思，应该改成“A的基数不是小于等于B的基数”.

8.5节，171页，习题8.5.20中，应当加上条件$\emptyset\not\in\Omega$.

10.4节,217页，习题10.4.1中，(b)里，$g(x)$应该为$g'(x)$.

18.2节，388页，第5行.“注意外测度对于每个集合都有定义，因为我们可以对任何非空集合取infimum”.这里译者把infimum放着没翻译，我认为翻译成“注意外测度对于每个集合都有定义，因为我们可以对任何非空集合取下确界”更好.但是译者也有他自己的理由，王先生说：

我的理解是：“下确界”只是对于有下界的集合而言，无下界的集合的infimum规定为“负无限”，它不是实数，不应称之为“下确界”。当然，如果硬规定它叫做“下确界”，也无话说。只不过说“无下界的集合的下确界是...”，不太像话。不知意下如何？

但是，我仍然认为应该把infimum译作“下确界”，这是因为陶哲轩在这里讨论的应该是广义实数，对于广义实数集合(广义实数集就是普通的实数集再添进$+\infty$和$-\infty$两个元素)来说，总是有下确界的.更详细的资料请读者见《陶哲轩实分析》中关于广义实数系的那一节，以自行判断.

18.5节，401页，引理18.5.3中，“设$\Omega$是$\mathbb{R}^n$的可测集”应该改为“设$\Omega$是$\mathbb{R}^n$的可测子集”.

18.5节，401页，推论18.5.4中，“其中$f_ j:\Omega\to \mathbb{R}^m$ 是$f$的第$j$个坐标”应该改成“其中$f_ j:\Omega\to \mathbb{R}$是$f$的第$j$个坐标”.

另外，陶哲轩本人在他的博文中也发布了英文原版的勘误，其中有一些勘误中文版可能没有改过来，请读者亲自看他的博文,可能需要FQ.
http://terrytao.wordpress.com/books/analysis-i/ 和 http://terrytao.wordpress.com/books/analysis-ii/

更新1(2013.1.30):为了照顾不会.翻.墙.的读者,我把陶哲轩本人的勘误网页转过来了.

Analysis I

Analysis I

Last updated: Oct 14, 2012

Analysis, Volume I
Terence Tao
Hindustan Book Agency, January 2006
Paper cover, 422 pages. ISBN 81-85931-62-3

This is basically an expanded and cleaned up version of my lecture notes for Math 131A. 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 II. It is currently in its second edition.

There are no solution guides for this text.

Sample chapters (contents, natural numbers, set theory, integers and rationals, logic, decimal system, index)

— Errata —

p. 2, item 3: “can you add” should be “Can you add”.
p. 9, line 5: “right-hand side” should be “left-hand side”.
p. 10, first display: $\frac{\partial^2}{\partial x \partial y}$ should be $\frac{\partial^2}{\partial y \partial x}$ .
p. 5, line 6 from bottom: $\sin(\pi/2-2)$ should be $\sin(\pi/2-z)$ . (Actually, for pedagogical reasons, it may be slightly better to use $\pi/2+z$ throughout this example instead of $\pi/2-z$ .)
p. 59, Lemma 3.3.12: f should map Z to W, and h should map X to Y. In the proof of this lemma (on page 60): $g \circ h$ is a function from X to Z, and $f \circ g$ is a function from Y to W.
p. 67, last paragraph: $\alpha \in A$ should be $\alpha \in I$ .
p. 98: In Exercise 4.2.1, Corollary 2.3.7 should be Corollary 4.1.9. In Exercise 4.2.6, $x,y,z$ should be rational numbers, not real.
p. 101: In Definition 4.3.9, after “ $x^0 := 1$ “, add “; in particular, we define $0^0 := 1$ “.
p. 127: In Exercise 5.3.4: add “(Hint: use Exercise 5.2.2.)”.
p. 131, line 12 from bottom: “they cannot be than” should be “they cannot be larger than”.
p. 175, Exercise 6.6.3: In the hint, replace “introduce” by “recursively introduce”, and insert “; $n > n_{j-1}$ ” after “ $|a_n| \geq j$ ” (two occurrences), with the parenthetical “(omitting the $n > n_{j-1}$ condition when $j=0$ )” inserted after the recursive definition of $n_j$ .
p. 197, in second line of proof of Proposition 7.3.4: the second sum should be $\sum_{k=0}^\infty$ rather than $\sum_{k=0}^K$ .
p. 216, Exercise 8.1.9: It needs to be noted that this exercise requires the axiom of choice from Section 8.4.
p. 220, Lemma 8.2.5: It needs to be noted that this lemma requires the axiom of choice from Section 8.4. Similarly, the case in Proposition 8.2.6 in which X is uncountable requires the axiom of choice also.
p. 227, Exercise 8.3.2: $g(x) := f(x)$ should be $g(x) := f^{-1}(x)$ .
p. 236, last line: “for any good set Y’” should be “for any good set Y’ with $A \cap Y'$ non-empty”.
p. 255, Proposition 9.3.9(b): $f(x_0)$ should be $L$ .
p. 303, Exercise 10.4.3(a): The limit should be in the set $(0,\infty) \backslash \{1\}$ rather than $(0,\infty)$ .
p. 336, line 13: replace “we have made no assumption on $\alpha$ ” with “the function $\alpha: {\Bbb R} \to {\Bbb R}$ could have been arbitrary”.
p. 337, Exercise 11.8.1: Lemma 11.8.1 should be Lemma 11.8.4.
p. 337, Exercise 11.8.5: In the last display, $f(0)$ should be $2f(0)$ .
p. 342, Exercise 11.9.1: “the function f is not differentiable” should be “the function $F(x) := \int_{[0,x]} f$ is not differentiable.
p. 383, first display: $a_n \times \hbox{ten}^i$ should be $a_n \times \hbox{ten}^n$ .
p. 387, fourth display: $a_n$ should be $a_{n+1}$ .

— Errata for the second edition (hardback) —

p. xii, bottom: “solidifed” –> “solidified”.
p. xiv, top: “to know how to to” –> “to know how to”.
p. 19. In footnote 2, add: “In the converse direction, if we have $n=m$ , then we may deduce $n++=m++$ ; this is the axiom of substitution (see Appendix A.7) applied to the operation $++$ .”
p. 24, after Definition2.2.1: “defined $n+m$ for every integer $n$ ” should be “defined $n+m$ for every natural number $n$ “.
p. 26, after Proposition 2.2.6: ”these notes” should be “this text”.
p. 28, Proposition 2.2.14: “and Let” should be “and let”.
p. 30, Lemma 2.3.3: “Natural numbers have no zero divisors” should read “Positive natural numbers have no zero divisors”.
p. 32, Definition 2.3.11: Add the remark “In particular, we define $0^0$ to equal $1$ .”
p. 37, Example 3.1.10: “(why?)” should be “(why?))”.
p. 45: “8-m, where n is a…” should be “8-m, where m is a…”. In Exercise 3.1.2, add Axiom 3.1 to the list of permitted axioms. In Exercise 3.1.1: (3.1.4) should be Definition 3.1.4.
p. 50: In the first line, $h(2n+3)=h(2n+2)$ should be $h(2n+3)=2n+2$ , and $N \backslash \{0\}$ should be ${\bf N} \backslash \{0\}$ .
p. 55, Exercise 3.3.1: $f \circ g$ and $\tilde f \circ \tilde g$ should be $g \circ f$ and $\tilde g \circ \tilde f$ respectively.
p. 61: In Exercise 3.4.8, Axiom 3.1 should be added to the list of permitted axioms.
p. 64: In Example 3.5.9, ” $(x_2,x_3) \in X_3$ ” should be “ $(x_2,x_3) \in X_2 \times X_3$ “.
p. 70, 4th line of proof of Lemma 3.6.9: $1 \leq i \leq N$ should be $1 \leq i \leq n$ . In the 6th line of proof of Proposition 3.6.8: Proposition 3.6.4 should be Lemma 3.6.9. After Lemma 3.6.9, add the following remark: “Strictly speaking, the expression $n-1$ has not yet been defined. For the purposes of this lemma, we temporarily define it to be the unique natural number $m$ such that $m++=n$ (which exists and is unique by Lemma 2.2.10).”
p. 81, before Lemma 4.2.3: ”product of a rational number” -> “product of two rational numbers”.
p. 84, before Definition 4.2.6: a space is missing between “Proposition 4.2.4″ and “allows”. Before this paragraph, add “In a similar spirit, we define subtraction on the rationals by the formula $x-y := x + (-y)$ , just as we did with the integers.”
p. 86: In Definition 4.3.2, “real numbers” should be “rational numbers”. In definition 4.3.4, “be a rational number” should be added after “Let $\varepsilon>0$ “.
p. 88: In Proposition 4.3.10(b), the hypothesis n>0 should be added.
p. 104, proof of Lemma 5.3.7; after invoking Proposition 4.3.7, add “(extended in the obvious manner to the $\delta=0$ case)”.
p. 105, after Proposition 5.3.10: $\lim_{n \to\infty} a_n$ should be $\hbox{LIM}_{n \to \infty} a_n$ .
p. 108, proof of Lemma 5.3.15: $n \geq N$ should be $n, m \geq N$ . ”This shows that $|a_n-a_m| \leq \varepsilon$ ” should read “This shows that $|a_n^{-1}-a_m^{-1}| \leq \varepsilon$ “.
p. 115: In the hint for Exercise 5.4.8, add “or Corollary 5.4.10″ after “use Proposition 5.4.9″.
p. 120: Add an additional exercise, Exercise 5.5.5: ”Establish an analogue of Proposition 5.4.14, in which “rational” is replaced by “irrational”.”
p. 124, Exercise 5.6.3: Add the hypothesis that x is non-zero (since the roots of 0 are not yet defined).
p. 126, proof of Proposition 6.1.4: Proposition 5.4.14 should be Proposition 5.4.12.
p. 134: In Definition 6.2.6(c) (and also on the first line of p. 135), $E - \{-\infty\}$ should be $E \backslash \{-\infty\}$ .
p. 135, Theorem 6.2.11(b), (c): Replace “Suppose that $M$ ” with “Suppose that $M \in {\Bbb R}^*$ ” (two occurrences). Exercise 6.2.2: Proposition 6.2.11 should be Theorem 6.2.11.
p.144: Cor. 6.4.14: line 4: ” .. for all $n \geq M$ ” should be ” .. for all $n \geq m$ “
p. ???: proof of Theorem 6.4.18: Replace “from Corollary 6.1.17″ here by “from Lemma 5.1.15 (or more precisely, the extension of that lemma to the real numbers, which is proven in exactly the same fashion)”.
p. 151, Exercise 6.6.5: Replace “the formula $n_j := \min\{n \in {\Bbb N}: |a_n-L| \leq 1/j\}$ , explaining why the set $\{n \in {\Bbb N}: |a_n-L| \leq 1/j\}$ is non-empty” with “the recursive formula $n_j := \min\{n > n_{j-1}: |a_n-L| \leq 1/j\}$ , with the convention $n_0=0$ , explaining why the set $\{n > n_{j-1}: |a_n-L| \leq 1/j\}$ is non-empty”.
p. 164, Definition 7.2.2: $(S_N)_{n=m}^\infty$ should be $(S_N)_{N=m}^\infty$ .
p. 169, Exercise 7.2.6: Add “How does the proposition change if we assume that $a_n$ does not converge to zero, but instead converges to some other real number $L$ ?”. After Corollary 7.3.2: “conditionally divergent” should be “not conditionally convergent”.
p. 176: “absolutely divergent series” should be “series that is not absolutely convergent”.
p. 177, Theorem 7.5.1: “conditionally divergent” should be “not conditionally convergent”, and similarly “absolutely divergent” should be “not absolutely convergent”. Similarly for Corollary 7.5.3 on page 179.
p. 186, Exercise 8.1.1: This exercise requires the axiom of choice, Axiom 8.1. In Exercise 8.1.4. $f(0), f(1), \ldots, f(n)$ should be $f(0), f(1),\ldots,f(n-1)$ .
p. 192, proof of Theorem 8.2.8: “absolutely divergent” should be “not absolutely convergent” (two occurrences).
p. 196, Remark 8.3.6: “Paul Cohen (1934-)” should now be “Paul Cohen (1934-2007)”. :-(
p. 197, Exercise 8.3.2: $f$ should be an injection rather than a bijection. In the definition of $g$ , $\bigcup_{n=0}^\infty D_n$ should be $\bigcup_{n=1}^\infty D_n$ (two occurrences).
p. 200, Exercise 8.4.1: $y \in y$ should be $y \in Y$ .
p. 206, Exercise 8.5.5: “ $f(x) \leq_Y f(x')$ ” should be “ $f(x) <_Y f(x')$ or $x=x'$ “. In Exercise 8.5.12, $x \leq_X x'$ should be $x <_X x'$ .
p. 208, Exercise 8.5.19: $Y := \{y \in Y': y < x \}$ should be $Y = \{ y \in Y': y <' x \}$ . In Exercise 8.5.20, the additional hypothesis “Assume that $\Omega$ does not contain the empty set $\emptyset$ ” should be added.
p. 214, Lemma 9.1.21. One needs the additional hypothesis “We assume that $a<b$ .”
p. 220, Definition 9.3.6: “ $f$ is $\varepsilon$ -close to $L$ near $x_0$ ” should be “ $f$ , after restricting to $E$ , is $\varepsilon$ -close to $L$ near $x_0$ “.
p. 228, Proposition 9.4.7: change “three items” to “four items”, and add “(d): For every $\varepsilon > 0$ , there exists a $\delta > 0$ such that $|f(x)-f(x_0)| \leq \varepsilon$ for all $x \in X$ with $|x-x_0| \leq \delta$ .
p. 232, proof of Proposition 9.5.3: after “Proposition 9.4.7″, add “(applied to the restriction of $f$ to the subdomain $X \cap (x_0,+\infty)$ )”.
p. 252, Proposition 10.1.7: One needs the additional hypothesis $x_0 \in X$ . Similarly for Proposition 10.1.10, Theorem 10.1.13, and Proposition 10.3.1.
p. 253, Definition 10.1.11: “For every $x_0 \in X$ ” should be “For every limit point $x_0 \in X$ “.
p. 254, Remark 10.1.14: Leibnitz should be Leibniz (two occurrences).
p. 256, Exercise 10.1.1: “ $x_0$ is also limit point of $Y$ ” should be “ $x_0 \in Y$ , and $x_0$ is also a limit point of $Y$ “.
p. 257, Definition 10.2.1: $x \in X$ should be $x_0 \in X$ .
p. ???: In the proof of Theorem 10.4.2,” $x_n = f^{-1}(y_0)$ ” should be “ $x_n = f^{-1}(y_n)$ “.
p. 271, Remark 11.2.2: “constant on $f$ ” should be “constant on $E$ “.
p. 290: In the proof of Proposition 11.7.1, in the third display, $[0,1]$ should be $|[0,1]|$ .

Note that the first edition paperback page numbers differ from the second edition hardback page numbers, which should be born in mind when applying the second edition errata to the first edition. (The section, theorem and exercise numbering, however, is mostly unchanged.)

Thanks to Tai-Danae Bradley, Brian, Eduardo Buscicchio, Evangelos Georgiadis, Ulrich Groh, Erik Koelink, Matthis Lehmkühler, Percy Li, Ming Li, Manoranjan Majji, Pieter Naaijkens, Vineet Nair, Cristina Pereyra, David Radnell, Tim Reijnders, Pieter Roffelsen, Luke Rogers, Marc Schoolderman, Kent Van Vels, Daan Wanrooy, Yandong Xiao, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections.

Analysis II

Analysis II

Last updated November 1, 2012

Analysis, Volume II
Terence Tao
Hindustan Book Agency, January 2006
Paper cover, 274 pages. ISBN 81-85931-62-3

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. 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 $a, b \leq X$ 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”. In Exercises 13.5.6 and 13.5.7, “Hausdorff” should be “not Hausdorff”.
p. ???: 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 “ $d_m (x-b)^m$ .
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. 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”.

Caution: the page numbering is not consistent across editions.

Thanks to Biswaranjan Behera, Carlos, EO, Florian, Gökhan Güçlü, Bart Kleijngeld, Eric Koelink, Wang Kunyang, Matthis Lehmkühler, Jason M., Manoranjan Majji, Geoff Mess, Cristina Pereyra, Kent Van Vels, Haokun Xu, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections.