Page 192 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 16: Continuous Linear Transformations

Notes

Note
We may find many K’s satisfying the above condition for a given bounded function. If it
is satisfied for one K, it is satisfied for a K > K.
1
Theorem 3: Let f be a linear functional defined on a normed linear space N, then f is bounded
f is continuous.

Proof: Let us first show that continuity of f boundedness of f.
If possible let f is continuous but not bounded. Therefore, for any natural number n, however
large, there is some point x such that
n
|f (x )| n || x || … (1)
n n
x
Consider the vector, y = n so that
n n x n
1
y = .
n
n
y 0 as n
n
y 0 in the norm.
n
Since any continuous functional maps zero vector into zero and f is continuous f (y ) f (0) = 0.
n
1
But |f (y )| = f (x ) … (2)
n n x n
n
It now follows from (1) & (2) that |f (y )| > 1, a contradiction to the fact that f (y ) 0 as n .
n n
Thus if f is bounded, then f is continuous.
Conversely, let f is bounded. Then for any sequence (x ), we have
n
|f (x )| K || x ||  n = 1, 2, …, and K 0.
n n
Let x 0 as n then
n
f (x ) 0 f is continuous at the origin and consequently it is continuous everywhere.
n
This completes the proof of the theorem.

Note The set of all bounded linear function on N is a vector space denoted by N*. As in the
case of linear operators, we make it a normed linear space by suitably defining a norm of
a functional f.

16.1.3 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 … (1)
x 0 x

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