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Mathematics Department Stanford University
Math 175 Homework 2
1. Suppose
X
is a Banach space (i.e. a complete normed linear space) and
S
is a linear
subspace of
X
. Prove that
S
, viewed as a normed linear space with the norm of
X
, is
complete if and only if
S
, viewed as a subset of
X
, is closed.
2. Let
X
be a normed linear space and
S
a linear subspace with
S
¤
X
. Prove that
S
contains no ball of
X
.
3. Let
M
¤
∅
be a metric space (not necessarily a linear space) with metric
d
.
(i) Given an open ball
B
±
.x/
, an open dense subset
U
of
M
, prove there is a ball
B
²
.y/
±
B
±
.x/
\
U
. (Note: Recall
U
dense
means
U
D
M
.)
(ii) If
U
1
;U
2
;:::
are open dense sets in
M
, prove there exist a “nested” sequence of open
balls
B
±
1
.x
1
/
²
B
±
2
.x
2
/
² ³³³
such that
±
j
<
1
=j
and
B
2
±
j
.x
j
/
±
U
j
for each
j
D
1
;
2
;:::
.
Hint: (i) already provides the inductive step for this construction.
4. Let
M
¤
∅
be a complete metric space. Prove
(i) If
U
1
;U
2
;:::
are open and dense, then
\
1
j
D
1
U
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This note was uploaded on 02/01/2011 for the course MATH 171 taught by Professor R during the Fall '09 term at Stanford.
 Fall '09
 R
 Math

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