Page 99 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
1
n p
Now x x e |x | p … (2)
k k k
k 1 k n 1
The R.H.S. of (2) gives the remainder after n terms of a convergent series (1).

1
p
Hence |x | p 0 as n . … (3)
k
k n 1
From (2) and (3) if follows that

x = x e … (4)
k k
k 1
n
Let f  * and S = x e then
p n k k
k 1
S x as n (Using (4))
n
Since f is linear, we have

n
f (S ) = x f(e ) .
n k k
k 1
Also f is continuous and S x, we have
n
f (S ) f (x) as n .
n
n
f (x) = x f(e ) … (5)
k k
k 1
which gives the form of the functional on  .
p
Now we establish the isomeric isomorphism of  * onto  , for which proceed as follows:
q
p
Let f (e ) = and show that the mapping
k k
T :  *  given by
q
p
T (f) = ( , , …, , …) is an isomeric isomorphism of  * onto  .
q
1 2 k p
First, we show that T is well defined.
For let x  , where x = ( , , …, , 0, 0, …) where
p
2
n
1
| | g 1 sgn k 1 k n
,
= k
k 0 n k
q–1
| | = | | for 1 k n.
k k
p 1 1
(q 1)
| | = | | = | | .  q p(q 1) q
q
p
k k k
p q
92 LOVELY PROFESSIONAL UNIVERSITY