Page 203 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
Let x  so that
p
x = x e k .
k
k 1
We shall show that

g (x) = x k k is the required g.
k 1
Since the representation for x is unique, g is well defined and moreover it is linear on  . To
p
prove it is bounded, consider
n
|g (x) | = k x k k x k
k 1 k 1
1 1
p q
p q
x k k (Using Hölder’s inequality)
k 1 k 1
1
q
q
|g (x)| x k .
k 1
g is bounded linear functional on  .
p
since e  for k = 1, 2, …, we get
k p
g (e ) = for any k so that
k k
*
T = ( ) and T is on  onto  .
p
g k p
We next show that
Tf f so that T is an isometry.

Since Tf  , we have from (6) and (10) that
q
1
q
q
k = || Tf || || f || … (12)
k 1

Also, x  p x x e .
k Hence
k
k 1
f (x) = x (e ) x k k .
k
k
k 1 k 1
|f (x)| x k k
k 1

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