-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
γ
∈S is an upper bound of E, then
γ
≥
α
’’ 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:
If
α
were not a lower bound of B, there would be some x∈B satisfying x<
α
. This x would be
an upper bound of L (by the preceding paragraph), contradicting our assumption that
α
is the least
upper bound of L.So
α
is a lower bound of B. Now if y is any lower bound of B, then y∈L,so
y ≤ sup L =
α
; this shows that
α
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
α
} with all G
α
∈T one has ∪
α
G
α
∈T.
(ii) For any finite collection {G
α
} with all G
α
∈T one has ∩
α
G
α
∈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)
资源评论
zalois2024-08-20这只是第三版的勘误吗？
wubia1112014-06-11这个是第三版的勘误
j2recover2015-12-17非常经典的书
一碗小笼2014-12-14的确是勘误，不是书不过也挺好的，原版的这个网站里也有貌似- baker20082014-11-04勘误，和第3版配合使用。
