Page 206 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 16: Continuous Linear Transformations

 If f is a bounded linear functional on a normed space N, then the norm of f is defined as: Notes

f(x)
f = sup
x 0 x

16.3 Keywords

Bounded Linear Functional: A linear functional on a normed linear space N is said to be bounded,
if there exists a constant k such that

f (x) K x x N .
Continuous Linear Transformations: Let N be a normed linear space. Then we know the set R of
real numbers and the set C of complex numbers are Banach spaces with the norm of any x R or
x C given by the absolute value of x. Thus with our previous notations, (N, R) or (N, C)
denote respectively the set of all continuous linear transformations from N into R or C.
Norm of a Bounded Linear Functional: If f is a bounded linear functional on a normed space N,
then the norm of f is defined as:

f(x)
|| f|| = sup
x 0 x
Second Conjugate: The conjugate space (N*)* of N* is called the second conjugate space of N .

16.4 Review Questions

1. Prove that the conjugate space of  is  ,
1
i.e.  *  .
1
2. Prove that the conjugate space of c is  .
o 1
or c * 
o 1
1 1
3. Let p > 1 with = 1 and let g L (X).
p q q
Then prove that the function defined by
F (f) = fg d for f Lp (X)
X
is a bounded linear functional on Lp (X) and

F = g
q
4. Let N be any n dimensional normed linear space with a basis B = {x , x , ..., x }. If
1 2 n
(r , r , ..., r ) is any ordered set of scalars, then prove that, there exists a unique continuous
1 2 n
linear functional f on N such that
f (x ) = r for i = 1, 2, …, n
i i
5. If T is a continuous linear transformation of a normed linear space N into a normed linear
space N , and if M is its null space, then show that T induces a natural linear transformation
T of N/M into N and that T = T .

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