Page 102 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

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

Notes
|f (x)| |x || k |
k
k 1
1 1
n q p
| | q x p (Using Hölder’s inequality)
k k
k 1 k 1
1
n q
or |f (x)| | k | q x x  p .
k 1
Hence we have
1
|f(x)| q
sup | | q = Tf (Using (6))
x 0 x k
k 1
which upon using definition of norm yields
f Tf … (13)

Thus f = Tf (Using (12) and (13))
From the definition of T, it is linear. Also since it is an isometry, it is one-to-one and onto
(already shown). Hence T is an isometric isomorphism of  * onto  , i.e.,
q
p
 * p  .
q
This completes the proof of the theorem.

1 1
Theorem 5: Let p > 1 with 1 and let g L (X). Then the function defined by
p q p
F (f) = fg d for f L (X)
p
X
is a bounded linear functional on L (X) and
p
F = g … (1)
q
Proof: We first note that

F is linear on L (X). For if f , f L (X), then we get
p 1 2 p
F (f + f ) = (f f )g d f g d f g d
1 2 1 2 1 2
X X X
= F (f ) + F (f )
1 2
So that
F (f + f ) = F (f ) + F (f )
1 2 1 2
and F ( F) = fg d F(f) .
X

Now |F (f)| = fg d |fg|d … (2)
X X

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