Page 215 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes

Notes Let L be a linear space. A mapping p : L R is called a sub-linear functional on L if
it satisfies the following two properties namely,

(i) p (x + y) p (x) + p (y) x, y L (sub additivity)
(ii) p ( x) = p (x), 0 (positive homogeneity)
Thus p defined on L in the above theorems is a sub-linear functional on L.

Some Applications of the Hahn-Banach Theorem

Theorem: If N is a normed linear space and x N, x 0 then there exists a functional f N* such
o o o
that
f (x ) = x and f = 1.
o o o o
Proof: Let M denote the subspace of N spanned by x , i.e.,
o
M = { x : any scalar}.
o
Define f : M F (R or C) by

f ( x ) = x .
o o
We show that f is a functional on M with f = 1.
Let x , x M so that
1 2
x = x and x = x . Then
1 1 o 2 2 o
f (x + x ) = f ( x + x )
1 2 1 o 2 o
= ( + ) x
1 2 o
But ( + ) x = x + x
1 2 o 1 o 2 o
= f (x ) + f (x )
1 2
Hence f (x + x ) = f (x ) + f (x ) … (1)
1 2 1 2
Let k be a scalar (real or complex). Then if x M, then x = x .
o
Now f (kx) = f (k x ) = k x = k f (x) … (2)
o o
If follows from (1) and (2) that f is linear.
Further, we note that since x M with = 1, we get
o
f (x ) = x .
o o
For any x M, we get, | f (x)| = | | || x || = x = x
o o
|f (x) | = x

f is bounded and we have

|f(x)|
sup = 1 for x M and x 0.
x
So by definition of norm of a functional, we get
f = 1.

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