1
ERRATA AND ADDENDA TO CHAPTERS 17 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
≥
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
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
.S
o
is a lower bound of
B
. Now if
y
is any lower bound of
B
, then
y
∈
L
,s
o
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
;i
f
x
=
0
let
u
be any vector with 
u
.
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
.I
f
y∕x
< 0, take
n
= 1; otherwise, write
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
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 Fall '08
 Zhou,X
 Topology, Addition, Metric space, Open set, Topological space, Closed set

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