Final Exam, Math 245C, June 4, 2007.
Exams are due June 13 at 6:00 PM in my mailbox. You may use your course notes
or any book available, including Folland and H¨
ormander, but you should write out a
complete solution of each problem. J. Garnett
1. A topological space
X
with topology
T
is
metrizable
if there is a metric
d
on
X
such that
U
∈ T
if and only if for all
x
∈
U
there is
r >
0 such that
B
(
x,r
) =
{
y
∈
X
:
d
(
x,y
)
< r
} ⊂
U.
Let
X
be a compact Hausdor± space and let
C
(
X
) be the Banach space
of continuous realvalued functions on
X
with the norm

f

= sup
{
f
(
x
)

:
x
∈
X
}
.
Prove that
C
(
X
) is separable (i.e. has a countable dense subset) if and only if
X
is
metrizable.
2. Let
X
be a real Banach space, let
M
be a closed subspace of
X
and let
L
:
M
→
R
be a continuous linear functional on
M
with

L

M
*
= sup
{
L
(
x
)

:
x
∈
M,

x

= 1
}
= 1
.
Prove that there is a
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 Spring '10
 tao/analysis
 Math, Topology, Metric space, countable dense subset, compact Hausdorff space, U. Let

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