Page 196 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 16: Continuous Linear Transformations

We have from (1) by using the notation of the Zeroth norm in a finite dimensional space, Notes
n
| f (x)| x f(x ) … (2)
0 i
i 1

n
If f(x ) = M, then from (2), we have
i
i 1
| f (x)| M x .
0
Hence f is bounded with respect to .
0
Since any norm on N is equivalent to , f is bounded with respect to any norm on N.
0
Consequently, f is continuous on N.

16.1.5 Representation Theorems for Functionals

We shall prove, in this section, the representation theorems for functionals on some concrete
Banach spaces.
n
n
Theorem 4: If L is a linear space of all n-tuples, then (i)  *  .
p q
Proof: Let (e , e , …, e ) be a standard basis for L so that any x = (x , x , …, x ) L can be written
1 2 n 1 2 n
as
x = x e + x e + … + x e .
1 1 2 2 n n
If f is a scalar valued linear function defined on L, then we get
f (x) = x f (e ) + x f (e ) + … + x f (e ) … (1)
1 1 2 2 n n
f determines and is determined by n scalars
y = f (e ).
i i
Then the mapping
y = (y , y , …, y ) f
1 2 n
n
where f (x) = x y i is an isomorphism of L onto the linear space L of all function f. We shall
i
i 1
establish (i) – (iii) by using above given facts.
(i) If we consider the space
n
th
L =  (1 p < ) with the p norm, then f is continuous and L represents the set of all
p
n
continuous linear functionals on  so that
p
L =  n p * .
Now for y f as an isometric isomorphism we try to find the norm for y’s.

For 1 < p < , we show that
 n * =  .
n
p q

LOVELY PROFESSIONAL UNIVERSITY 189