Page 98 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

p
Unit 8: Bounded Linear Functional on the L -spaces

It follows now from (11) and (13) that Notes
n
i so that y
f = |y | f is an isometric isomorphism.
i 1
Hence,  n *  .
n
1
This completes the proof of the theorem.

Note We need the signum function for finding the conjugate spaces of some infinite
dimensional space which we define as follows:
If is a complex number, then

sgn if 0
| |
0 if 0

(i) |sgn | = 0 if = 0 and | sgn | = 1 if 0

(ii) sgn = 0 if = 0 and sgn = =| |, if 0.
| |

Theorem 4: The conjugate space of  is  , where
p q
1 1
= 1 and 1 < p < .
p q

or  * p  .
q

Proof: Let x = (x )  so that |x | p … (1)
n p n
n 1
th
Let e = (0, 0, 0, …, 1, 0, 0, …) where 1 is in the n place.
n
e  for n = 1, 2, 3, … .
p
n
*
We shall first determine the form of f and then establish the isometric isomorphism of  onto
p
 q .
By using (e ), we can write any sequence
n
n
(x , x , … x , 0, 0, 0, …) in the form x e and
1 2 n k k
k 1
n
x x e (0, 0, 0, , x , x , ).
k k n 1 n 2
k 1

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