HW102 - Analysis 1 Homework 02 1(i Let z c0 be nonzero nd c...

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