Page 96 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

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

y f is an isometric isomorphism. Notes
n
Hence  n p * =  .
q
n
(ii) Let L =  with the norm defined by
1
n
x = |x |
i
i 1
Now f defined in (1), above is continuous as in (i) and L here represents the set of continuous
n
linear functional on  so that
1
n
L =  * .
1
We now determine the norm of y’s which makes y f an isometric isomorphism.

n
Now, f (x) = x y
i i
i 1

n
|x ||y |
i
i
i 1
n n
But |x ||y | max. |y | |x | so that
i
i
i
i
i 1 i 1
n
f (x) max. |y | |x |.
i i
i 1
From the definition of the norm for f, we have
f = max. {|y | : i = 1, 2, …, n} … (8)
i
Now consider the vector defined as follows:

If |y | = max |y | , let us consider the vector x as
i 1 i n i

|y |
x = i when|y | max |y | and x 0 … (9)
i y i 1 i n i i
i
otherwise
From the definition, x = 0 k i, so that we have
k

y
(x) = i = 1
y

n
Further | f (x)| = (x y ) = |y |.
i
i
i
i 1
Hence |y | = |f (x)| f x
i

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