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《陶哲轩实分析》部分勘误

發(fā)布時(shí)間：2024/9/20 编程问答 27 豆豆

生活随笔收集整理的這篇文章主要介紹了《陶哲轩实分析》部分勘误小編覺得挺不錯(cuò)的,現(xiàn)在分享給大家,幫大家做個(gè)參考.

我在讀《陶哲軒實(shí)分析》，作者是陶哲軒,譯者王昆揚(yáng).2008年11月第一版，第一次印刷.我在此添加一部分中譯本印刷錯(cuò)誤，若網(wǎng)友發(fā)現(xiàn)了另外的錯(cuò)誤，請(qǐng)?jiān)谠u(píng)論里補(bǔ)充，由我代為添加.若有不當(dāng)之處，敬請(qǐng)指正.

勘誤：

3.2節(jié)，37頁(yè),公理3.9中，“不同”應(yīng)該改為“不交”.

3.4節(jié)，48頁(yè)，習(xí)題3.4.10中，$(\bigcap_{\alpha\in I}A_{\alpha})\bigcap(\bigcap_{\alpha\in J}A_{\alpha})=(\bigcap_{\alpha\in I\bigcap J}A_{\alpha})$應(yīng)當(dāng)改為$(\bigcap_{\alpha\in I}A_{\alpha})\bigcap(\bigcap_{\alpha\in J}A_{\alpha})=(\bigcap_{\alpha\in I\bigcup J}A_{\alpha})$

7.1節(jié)，130頁(yè)，命題7.1.11，(e),$f:X\bigcap Y\to \mathbb{R}$應(yīng)該改為$f:X\bigcup Y\to\mathbb{R}$.

7.4節(jié)，145頁(yè)，習(xí)題7.4.1，$\sum_{m=0}^Ma_{f(m)}$應(yīng)當(dāng)改為$\sum_{m=0}^{\infty}a_{f(m)}$.

7.5節(jié)，145頁(yè)，(b):如果$\alpha>1$,那么級(jí)數(shù)$\sum_{n=m}^{\infty}a_n$是條件發(fā)散的（從而是絕對(duì)發(fā)散的）。我認(rèn)為最好把（從而是絕對(duì)發(fā)散的）改為（從而不是絕對(duì)收斂的），后者似乎更符合原文.

8.5節(jié)，171頁(yè)，習(xí)題8.5.15中，“A的基數(shù)不小于或等于B的基數(shù)”這句話有歧義.到底是“不小于”或等于呢，還是不“小于或等于”呢？結(jié)合題目意思，應(yīng)該改成“A的基數(shù)不是小于等于B的基數(shù)”.

8.5節(jié)，171頁(yè)，習(xí)題8.5.20中，應(yīng)當(dāng)加上條件$\emptyset\not\in\Omega$.

10.4節(jié),217頁(yè)，習(xí)題10.4.1中，(b)里，$g(x)$應(yīng)該為$g'(x)$.

18.2節(jié)，388頁(yè)，第5行.“注意外測(cè)度對(duì)于每個(gè)集合都有定義，因?yàn)槲覀兛梢詫?duì)任何非空集合取infimum”.這里譯者把infimum放著沒翻譯，我認(rèn)為翻譯成“注意外測(cè)度對(duì)于每個(gè)集合都有定義，因?yàn)槲覀兛梢詫?duì)任何非空集合取下確界”更好.但是譯者也有他自己的理由，王先生說：

我的理解是：“下確界”只是對(duì)于有下界的集合而言，無下界的集合的infimum規(guī)定為“負(fù)無限”，它不是實(shí)數(shù)，不應(yīng)稱之為“下確界”。當(dāng)然，如果硬規(guī)定它叫做“下確界”，也無話說。只不過說“無下界的集合的下確界是...”，不太像話。不知意下如何？

但是，我仍然認(rèn)為應(yīng)該把infimum譯作“下確界”，這是因?yàn)樘照苘幵谶@里討論的應(yīng)該是廣義實(shí)數(shù)，對(duì)于廣義實(shí)數(shù)集合(廣義實(shí)數(shù)集就是普通的實(shí)數(shù)集再添進(jìn)$+\infty$和$-\infty$兩個(gè)元素)來說，總是有下確界的.更詳細(xì)的資料請(qǐng)讀者見《陶哲軒實(shí)分析》中關(guān)于廣義實(shí)數(shù)系的那一節(jié)，以自行判斷.

18.5節(jié)，401頁(yè)，引理18.5.3中，“設(shè)$\Omega$是$\mathbb{R}^n$的可測(cè)集”應(yīng)該改為“設(shè)$\Omega$是$\mathbb{R}^n$的可測(cè)子集”.

18.5節(jié)，401頁(yè)，推論18.5.4中，“其中$f_ j:\Omega\to \mathbb{R}^m$?是$f$的第$j$個(gè)坐標(biāo)”應(yīng)該改成“其中$f_ j:\Omega\to \mathbb{R}$是$f$的第$j$個(gè)坐標(biāo)”.

另外，陶哲軒本人在他的博文中也發(fā)布了英文原版的勘誤，其中有一些勘誤中文版可能沒有改過來，請(qǐng)讀者親自看他的博文,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):為了照顧不會(huì).翻.墻.的讀者,我把陶哲軒本人的勘誤網(wǎng)頁(yè)轉(zhuǎn)過來了.

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 “l(fā)eft-hand side”.
p. 10, first display: should be .
p. 5, line 6 from bottom: should be . (Actually, for pedagogical reasons, it may be slightly better to use throughout this example instead of .)
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): is a function from X to Z, and is a function from Y to W.
p. 67, last paragraph: should be .
p. 98: In Exercise 4.2.1, Corollary 2.3.7 should be Corollary 4.1.9. In Exercise 4.2.6, should be rational numbers, not real.
p. 101: In Definition 4.3.9, after ““, add “; in particular, we define “.
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 “; ” after “” (two occurrences), with the parenthetical “(omitting the condition when )” inserted after the recursive definition of .
p. 197, in second line of proof of Proposition 7.3.4: the second sum should be rather than .
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: should be .
p. 236, last line: “for any good set Y’” should be “for any good set Y’ with non-empty”.
p. 255, Proposition 9.3.9(b): should be .
p. 303, Exercise 10.4.3(a): The limit should be in the set rather than .
p. 336, line 13: replace “we have made no assumption on ” with “the function 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, should be .
p. 342, Exercise 11.9.1: “the function f is not differentiable” should be “the function is not differentiable.
p. 383, first display: should be .
p. 387, fourth display: should be .

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

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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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: should be .
p. 393, example 12.1.9: should be .
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: should be .
p. 416, line 11: “” should be ““.
p. 419, line -2: In Exercise 12.5.15, = should be . Also, “that by counterexample” should be “by counterexample that”
p. 426, Exercise 13.2.9: should be 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 and .”
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: “” should be ““.
p. 458: Exercise 14.5.2 should be deleted and redirected to Exercise 14.2.2(c).
p. 459: In line 11, should be .
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, should be , 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, should be .
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, should be .
p. 482: In Exercise 15.2.5, the on the right-hand side should be .
p. 486: In second and third display, y should be in rather than .
p. 493: In Exercise 15.5.4, should be .
p. 501: In Theorem 17.7.2, “if is not invertible” should be ”if 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 and not just in ; insert the line “By working with one component of at a time, we may assume ” as the first line of the proof. Also, should be .
p. 557: In the second display, should be .
p. 560: In Exercise 17.6.1, add the hypothesis “and is continuous” before “, show that is a strict contraction”.
p. 561: In Exercise 17.6.3, change “which is a strict contraction” to “such that for all distinct in “. In Exercise 17.6.8, should be .
p. 562: In Theorem 17.7.2, should be .
p. 565, line -7: should be rather than .
p. 570, first display: all partial derivatives should have a – sign (not just the first one). Last paragraph: “Thus lies in W” should be “Thus lies in U”.
p. 571, second display: add “” at the end.
p. 584, Corollary 18.2.7: “” should be ““.
p. 599, Definition 18.5.9: should be .
p. 600: In Lemma 18.5.10, should be . In the second and fourth lines of the proof of this lemma, should be .
p. 616-617, Exercise 19.2.10: should be throughout.

— Errata to the second edition (hardcover) —

p. 372, In Case 1 of the proof of Theorem 12.5.8, all occurrences of “ should be 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 such that the “interval” ” should be replaced with “there exists a set which is an interval for some , a ray for some , the ray for some , or the whole space , 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 , then .”
p. 396: In Exercise 14.1.5, should be , and should be .
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 ” should be “for some “.
p. 431: In Exercise 15.2.8(e), “” should be “.
p. 433 (proof of Theorem 15.3.1): in the third display and in the next line should be
and respectively.
p. 473: In Exercise 16.5.4, should be .
p. 477: In Example 17.1.7, should be .
p. 486: In Definition 17.3.7, should be , and should be .
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, should equal rather than .
p. 499, proof of Lemma 17.6.6: After “ does indeed map to itself.”, add “The same argument shows that for a sufficiently small , maps the closed ball to itself. After “ is a strict contraction”, add “on , and hence on the complete space “.
p.502, proof of Theorem 17.7.2: “” should be ““.
p. 505, Section 17.8: should be . In the second paragraph, the function should be (for better compatibility with the discussion of the implicit function theorem).
p.508, proof of Theorem 17.8.1, “U is open and contains ” should be “U is open and contains “.
p. 515: In the display before Definition 18.2.4, should be . In Definition 18.2.4, should be .
p. 520: In Example 18.2.9, should be in the display.
p. 528, proof of Lemma 18.4.8: On the second line, “l(fā)et be any other measurable set” should be “l(fā)et 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 and y=0, equals -1 when and y=0, and equals zero otherwise”.

Caution: the page numbering is not consistent across editions.

Thanks to Biswaranjan Behera, Carlos, EO, Florian, G?khan Gü?lü, 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.

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