《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

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: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 5, line 6 from bottom: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ . (Actually, for pedagogical reasons, it may be slightly better to use $《陶哲轩实分析》部分勘误$ throughout this example instead of $《陶哲轩实分析》部分勘误$ .)
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): $《陶哲轩实分析》部分勘误$ is a function from X to Z, and $《陶哲轩实分析》部分勘误$ is a function from Y to W.
p. 67, last paragraph: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 98: In Exercise 4.2.1, Corollary 2.3.7 should be Corollary 4.1.9. In Exercise 4.2.6, $《陶哲轩实分析》部分勘误$ should be rational numbers, not real.
p. 101: In Definition 4.3.9, after “ $《陶哲轩实分析》部分勘误$ “, add “; in particular, we define $《陶哲轩实分析》部分勘误$ “.
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 “; $《陶哲轩实分析》部分勘误$ ” after “ $《陶哲轩实分析》部分勘误$ ” (two occurrences), with the parenthetical “(omitting the $《陶哲轩实分析》部分勘误$ condition when $《陶哲轩实分析》部分勘误$ )” inserted after the recursive definition of $《陶哲轩实分析》部分勘误$ .
p. 197, in second line of proof of Proposition 7.3.4: the second sum should be $《陶哲轩实分析》部分勘误$ rather than $《陶哲轩实分析》部分勘误$ .
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: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 236, last line: “for any good set Y’” should be “for any good set Y’ with $《陶哲轩实分析》部分勘误$ non-empty”.
p. 255, Proposition 9.3.9(b): $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 303, Exercise 10.4.3(a): The limit should be in the set $《陶哲轩实分析》部分勘误$ rather than $《陶哲轩实分析》部分勘误$ .
p. 336, line 13: replace “we have made no assumption on $《陶哲轩实分析》部分勘误$ ” with “the function $《陶哲轩实分析》部分勘误$ 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, $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 342, Exercise 11.9.1: “the function f is not differentiable” should be “the function $《陶哲轩实分析》部分勘误$ is not differentiable.
p. 383, first display: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 387, fourth display: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .

— 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 $《陶哲轩实分析》部分勘误$ , then we may deduce $《陶哲轩实分析》部分勘误$ ; this is the axiom of substitution (see Appendix A.7) applied to the operation $《陶哲轩实分析》部分勘误$ .”
p. 24, after Definition2.2.1: “defined $《陶哲轩实分析》部分勘误$ for every integer $《陶哲轩实分析》部分勘误$ ” should be “defined $《陶哲轩实分析》部分勘误$ for every natural number $《陶哲轩实分析》部分勘误$ “.
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 $《陶哲轩实分析》部分勘误$ to equal $《陶哲轩实分析》部分勘误$ .”
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, $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ , and $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 55, Exercise 3.3.1: $《陶哲轩实分析》部分勘误$ and $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ and $《陶哲轩实分析》部分勘误$ 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, ” $《陶哲轩实分析》部分勘误$ ” should be “ $《陶哲轩实分析》部分勘误$ “.
p. 70, 4th line of proof of Lemma 3.6.9: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ . 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 $《陶哲轩实分析》部分勘误$ has not yet been defined. For the purposes of this lemma, we temporarily define it to be the unique natural number $《陶哲轩实分析》部分勘误$ such that $《陶哲轩实分析》部分勘误$ (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 $《陶哲轩实分析》部分勘误$ , 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 $《陶哲轩实分析》部分勘误$ “.
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 $《陶哲轩实分析》部分勘误$ case)”.
p. 105, after Proposition 5.3.10: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 108, proof of Lemma 5.3.15: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ . ”This shows that $《陶哲轩实分析》部分勘误$ ” should read “This shows that $《陶哲轩实分析》部分勘误$ “.
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), $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 135, Theorem 6.2.11(b), (c): Replace “Suppose that $《陶哲轩实分析》部分勘误$ ” with “Suppose that $《陶哲轩实分析》部分勘误$ ” (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 $《陶哲轩实分析》部分勘误$ ” should be ” .. for all $《陶哲轩实分析》部分勘误$ “
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 $《陶哲轩实分析》部分勘误$ , explaining why the set $《陶哲轩实分析》部分勘误$ is non-empty” with “the recursive formula $《陶哲轩实分析》部分勘误$ , with the convention $《陶哲轩实分析》部分勘误$ , explaining why the set $《陶哲轩实分析》部分勘误$ is non-empty”.
p. 164, Definition 7.2.2: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 169, Exercise 7.2.6: Add “How does the proposition change if we assume that $《陶哲轩实分析》部分勘误$ does not converge to zero, but instead converges to some other real number $《陶哲轩实分析》部分勘误$ ?”. 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. $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
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: $《陶哲轩实分析》部分勘误$ should be an injection rather than a bijection. In the definition of $《陶哲轩实分析》部分勘误$ , $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ (two occurrences).
p. 200, Exercise 8.4.1: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 206, Exercise 8.5.5: “ $《陶哲轩实分析》部分勘误$ ” should be “ $《陶哲轩实分析》部分勘误$ or $《陶哲轩实分析》部分勘误$ “. In Exercise 8.5.12, $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. 208, Exercise 8.5.19: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ . In Exercise 8.5.20, the additional hypothesis “Assume that $《陶哲轩实分析》部分勘误$ does not contain the empty set $《陶哲轩实分析》部分勘误$ ” should be added.
p. 214, Lemma 9.1.21. One needs the additional hypothesis “We assume that $《陶哲轩实分析》部分勘误$ .”
p. 220, Definition 9.3.6: “ $《陶哲轩实分析》部分勘误$ is $《陶哲轩实分析》部分勘误$ -close to $《陶哲轩实分析》部分勘误$ near $《陶哲轩实分析》部分勘误$ ” should be “ $《陶哲轩实分析》部分勘误$ , after restricting to $《陶哲轩实分析》部分勘误$ , is $《陶哲轩实分析》部分勘误$ -close to $《陶哲轩实分析》部分勘误$ near $《陶哲轩实分析》部分勘误$ “.
p. 228, Proposition 9.4.7: change “three items” to “four items”, and add “(d): For every $《陶哲轩实分析》部分勘误$ , there exists a $《陶哲轩实分析》部分勘误$ such that $《陶哲轩实分析》部分勘误$ for all $《陶哲轩实分析》部分勘误$ with $《陶哲轩实分析》部分勘误$ .
p. 232, proof of Proposition 9.5.3: after “Proposition 9.4.7″, add “(applied to the restriction of $《陶哲轩实分析》部分勘误$ to the subdomain $《陶哲轩实分析》部分勘误$ )”.
p. 252, Proposition 10.1.7: One needs the additional hypothesis $《陶哲轩实分析》部分勘误$ . Similarly for Proposition 10.1.10, Theorem 10.1.13, and Proposition 10.3.1.
p. 253, Definition 10.1.11: “For every $《陶哲轩实分析》部分勘误$ ” should be “For every limit point $《陶哲轩实分析》部分勘误$ “.
p. 254, Remark 10.1.14: Leibnitz should be Leibniz (two occurrences).
p. 256, Exercise 10.1.1: “ $《陶哲轩实分析》部分勘误$ is also limit point of $《陶哲轩实分析》部分勘误$ ” should be “ $《陶哲轩实分析》部分勘误$ , and $《陶哲轩实分析》部分勘误$ is also a limit point of $《陶哲轩实分析》部分勘误$ “.
p. 257, Definition 10.2.1: $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .
p. ???: In the proof of Theorem 10.4.2,” $《陶哲轩实分析》部分勘误$ ” should be “ $《陶哲轩实分析》部分勘误$ “.
p. 271, Remark 11.2.2: “constant on $《陶哲轩实分析》部分勘误$ ” should be “constant on $《陶哲轩实分析》部分勘误$ “.
p. 290: In the proof of Proposition 11.7.1, in the third display, $《陶哲轩实分析》部分勘误$ should be $《陶哲轩实分析》部分勘误$ .

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

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