MATH 7338
HOMEWORK #2
Due date: October 8, 2009
Work the following problems and hand in your solutions.
You may work together with
other people in the class, but you must each write up your solutions independently. A subset
of these will be selected for grading. Write LEGIBLY on the FRONT side of the page only,
and STAPLE your pages together.
1.
(a) Let
δ
n
= (0
,
0
, . . . ,
0
,
1
,
0
,
0
, . . .
) and
δ
0
= (1
,
1
,
1
, . . .
)
.
Show that
{
δ
n
}
n
≥
0
is a
Schauder basis for
c.
Note: You can use the fact (shown in class) that
{
δ
n
}
n
≥
1
is a Schauder basis for
c
0
,
so this
part should be easy.
(b) Show that
c
*
is isometrically isomorphic to
ℓ
1
.
(c) Show that
c
and
c
0
are topologically isomorphic and further that the two bases
described above are
equivalent
in the sense that there exists a topological isomorphism
T
:
c
→
c
0
that maps the basis
{
δ
n
}
n
≥
0
for
c
onto the basis
{
δ
n
}
n
≥
1
for
c
0
.
(d) Show that if
x
∈
c
0
and
bardbl
x
bardbl
∞
= 1
,
then there exist
y
negationslash
=
z
∈
c
0
with
bardbl
y
bardbl
∞
=
bardbl
z
bardbl
∞
= 1
such that
x
= (
y
+
z
)
/
2
.
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 Fall '09
 HEIL
 Math, Topology, ek, Dual space, Borel measure

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