Page 201 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
1
n p
p
Now x x e k x k … (2)
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
p
Hence x k 0 as n … (3)
k n 1
From (2) and (3), it follows that

x = x e k . … (4)
k
k 1
n
*
Let f  and s n x e then
p
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 isometric isomorphism of  onto  , for which we proceed as follows:
p
q
Let f (e ) = and show that the mapping
k k
*
T :   given by … (6)
q
p
*
T (f) = ( , , …, , …) is an isometric isomorphism of  onto  .
1 2 k p q
First, we show that T is well defined.
For let x  , where x = ( , , …, , 0, 0, …)
p
1
2
n
g 1 sgn , 1 k n
where = k k
k 0 n k
| | = | | q – 1 for 1 k n.
k k
1 1
p 
(q 1)
| | = = | | . q p(q 1) q
q
p
k k k p q
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