Page 90 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

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

The smallest constant k for which (1) holds is called the norm of f, written f . Notes

|f(x)|
Thus f = sup : x 0 and x N 1 or equivalently
x
f = sup {|f (x)| : x X and x = 1}.

Also |f (x)| f x x N .
1
p
p
Definition: Let p R, p > 0. We define L = L [0, 1] to be the set of all real-valued functions on
[0, 1] such that
1
1 p
(i) f is measurable and (ii) f = |f| p < .
p
0

1
Note L or simply L denotes the class of measurable function f (x) which are also
L-integrable.

p
8.1.3 Bounded Linear Functional on L -spaces
If x  and f is bounded linear functional on  , then f has the unique representation of the
p
p
form as an infinite series

f (x) = x f(e )
k
k
k 1
8.1.4 Norm

*
The norm of f  is given by
p
1
q
f = |f(e )| q
k
k 1
Likewise in finite dimensional case, the bounded linear functionals are characterised by the
values they assume on the set e , k = 1, 2, 3, … .
k
8.1.5 Continuous Linear Functional

A linear functional f is continuous if given > 0 there exists > 0 so that
|f (x) – f (y)| whenever x – y .

8.1.6 Theorems

1 1
p
Theorem 1: Suppose 1 p < , and 1 , then, with  = L we have
p q
q
* = L ,

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