Mathematics Department Stanford University
Math 175 Homework 4
1. Suppose that
H
is a Hilbert space and
{
x
k
}
k
D
1
;
2
;:::
is a sequence in
H
with the property that
{
.x
k
;x/
}
k
D
1
;
2
;:::
is bounded for each given
x
2
H
.
(i) If
S
j
D
{
x
2
H
W j
.x
k
;x/
j ±
j
8
k
D
1
;
2
;:::
}
,
j
D
1
;
2
;:::
, prove that each
S
j
is a closed convex
subset of
H
and
[
1
j
D
1
S
j
D
H
.
(ii) If the sets
S
j
are as in (i), prove that some
S
j
contains a ball.
Hint: hw2, Q.4
2. Suppose that
H
is a Hilbert space and
{
x
k
}
k
D
1
;
2
;:::
is a sequence in
H
with the property that
{
.x
k
;x/
}
k
D
1
;
2
;:::
is bounded for each given
x
2
H
. Prove that
{
x
k
}
k
D
1
;
2
;:::
is a bounded
sequence
in
H
(i.e. there is
M >
0 such that
k
x
k
k
< M
8
k
). (!)
Note: This property is called the uniform boundedness principle.
3. Let
X
be a complex normed linear space and suppose that
f
W
X
!
C
is linear (“
f
is a linear
functional on
X
”).
(i) Prove that ker
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 Fall '09
 R
 Math, Metric space, Hilbert space, Compact space, Topological vector space

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