Analysis 1
Homework 02
1. (i) Let
z
∈
c
0
be nonzero; find
φ
∈
c
*
0
with
k
φ
k
= 1 and
φ
(
z
) =
k
z
k
.
(ii) Let
z
∈
‘
1
be nonzero; find
φ
∈
‘
*
1
with
k
φ
k
= 1 and
φ
(
z
) =
k
z
k
.
2. For
z
= (
z
n
)
∞
n
=1
∈
c
0
write
φ
(
z
) =
∞
X
n
=1
z
n
2
n
.
Prove:
(i)
φ
is a bounded linear functional of norm 1;
(ii) if
z
∈
c
0
is nonzero then
|
φ
(
z
)
|
<
k
z
k
.
Solutions
1. (i) As
z
∈
c
0
there exists
N
such that
|
z
N
|
=
k
z
k
∞
>
0.
The specific
‘coordinate’ linear functional
φ
N
:
c
0
→
:
w
7→
w
N
clearly does the job in this
case: if
w
∈
c
0
then
|
φ
N
(
w
)
|
=
|
w
N
|
6
k
w
k
∞
while
|
φ
N
(
z
)
|
=
|
z
N
|
=
k
z
k
∞
.
(ii) Not quite so straightforward.
Seek:
a
∈
‘
∞
such that
k
a
k
∞
= 1 and
T
a
(
z
) =
k
z
k
1
- that is,
∑
n
∈
N
a
n
z
n
=
∑
n
∈
N
|
z
n
|
. Found: for each
n
let
a
n
be
any scalar of absolute value 1 such that
a
n
z
n
=
|
z

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- Spring '09
- LARSON
- Linear Algebra, Vector Space, Zn, Topological vector space, Linear functional
-
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