WalterRudin，PrinciplesofMathematicalAnalysis,3rdEdition

5星 · 超过95%的资源需积分: 32 141 浏览量更新于2023-07-09 评论收藏 29KB PDF 举报

身份认证购VIP最低享 7 折!

领优惠券(最高得80元）

资源详情

资源评论

资源推荐

-1-

ERRATA AND ADDENDA TO CHAPTERS 1-7 OF RUDIN’S

PRINCIPLES OF MATHEMATICAL ANALYSIS, 3rd Edition, 4th Printing (noted as of August, 2003)

For additional errata to earlier printings, see last page of these sheets.



Note: If you don’t want to write corrections into your text, you might put them on PostIts (or

slivers of paper cut from PostIts) and insert these at the page in question.



P.4, line 4: Change this line to ‘‘(ii) If

∈S is an upper bound of E, then

≥

’’ for greater clarity

P.4 3rd line of Deﬁnition 1.10: A clearer statement would be, ‘‘Every subset E ⊂ S which is nonempty

and bounded above has a supremum sup E in S.’’

P.5, last 5 lines of proof of Theorem 1.11 : Change these lines to:

were not a lower bound of B, there would be some x∈B satisfying x<

. This x would be

an upper bound of L (by the preceding paragraph), contradicting our assumption that

is the least

upper bound of L.So

is a lower bound of B. Now if y is any lower bound of B, then y∈L,so

y ≤ sup L =

; this shows that

is the greatest lower bound of B.

P.6, Proposition 1.14: Add

(e)–(x+y)=(–x)+(–y).

Can you see how to prove this? I will either discuss it in class, or make it an exercise.

P.12, deﬁnitions of operations on the extended real numbers: Rudin should have noted the convention that

x+(+

∞

) and x+(–

∞

) may be abbreviated x+

∞

and x–

∞

respectively, and mentioned that addition

and multiplication are understood to be commutative on the extended reals, so that the deﬁnitions he gives

also imply further cases like +

∞

+x =+

∞

. Finally, the three equations in (a), instead of having the

common condition ‘‘If x is real’’, should be preceded by the respective conditions, ‘‘If x is real or

∞

’’, ‘‘If x is real or –

∞

’’, and only in the last case simply ‘‘If x is real’’.

P.16, Theorem 1.37: Add one more part:

(g) Assuming k > 0, there exists a vector u with | u |=1 such that u·x =|x |.

Proof: If x ≠ 0 let u =|x|

–1

x;ifx = 0 let u be any vector with | u |=1.

P.19, middle: The author refers to the archimedean property of Q. This is not a consequence of

Theorem 1.20(a); that would be circular reasoning. Rather, it is an elementary property of Q: Given x,

y∈Q with x > 0, we need to ﬁnd an n>y⁄x.Ify⁄x< 0, take n = 1; otherwise, write y⁄x as a

fraction with positive denominator, and take for n any integer greater than its numerator.

P.36: After ﬁnishing the section of metric spaces, you might ﬁnd the following discussion enlightening;

but it is not required reading.

What is topology? Chapter 2 of Rudin is entitled ‘‘Basic Topology’’, but the chapter is about metric

spaces, and the word ‘‘topology’’ does not appear in that chapter, nor in the index. What does it refer to?

Topology is a ﬁeld of mathematics that includes the study of metric spaces as a special case. The key

to the connection between metric spaces and the more general concept of a topological space is

Theorem 2.24, parts (a) and (b) (p.34), which show that if we write T for the set of all open sets in a

metric space, then the union of any family of members of T, and the intersection of any ﬁnite family of

members of T, are also open sets. Families of sets with these properties come up in other contexts as

well; so one makes

Deﬁnition. A topological space X means a pair (X, T), where X is a set, and T is a set of subsets of

X which satisﬁes

(i) For any collection {G

} with all G

∈T one has ∪

∈T.

(ii) For any ﬁnite collection {G

} with all G

∈T one has ∩

∈T.

(iii) ∅∈T and X∈T.

When T has been speciﬁed, and there is no danger of ambiguity, one simply speaks of ‘‘the

topological space X’’. The sets in T are called the open sets of X.

(Remark: Condition (iii) can be omitted from this deﬁnition if one interprets conditions (i) and (ii)

本内容试读结束，登录后可阅读更多

下载后可阅读完整内容，剩余3页未读，立即下载

wubia111

2014-06-11

这个是第三版的勘误

mtguo

粉丝: 0
资源: 2

会员权益专享

Walter Rudin，Principles of Mathematical Analysis, 3rd Edition

评论4

会员权益专享

最新资源

Walter Rudin，Principles of Mathematical Analysis, 3rd Edition

评论4

Principles of mathematical analysis by Walter Rudin

数学分析Rudin Principles of Mathematical Analysis

walter rudin principles of mathematical analysis(third edition)

rudin principles of mathematical analysis 下载

principles of mathematical analysis 3rd

walter rudin，《数学分析原理》

杨力华编著的《实变函数论教程》都引用了哪些英文文献

你尝试去学习他们的证明BSD路线，然后根据他们的研究思路去找到BSD猜想的一个证明，并把最后的证明结果发给我

function analysis rudin电子版

数学分析原理rudin油管

rudin数学分析原理2册pdf

rudin数学分析原理 csdn下载

数学分析原理rudin完整答案

数学分析原理rudin pdf

数学分析原理rudin第一册pdf

rudin实分析第三版答案

实分析第四版rodyen中文答案

全变分模型 matlab

会员权益专享

最新资源