Page 191 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes

Note The conjugate space (N*)* of N* is called the second conjugate space of N and shall
be denoted by N**. Also note that N** is complete too.

Theorem 1: The conjugate space N* is always a Banach space under the norm

f(x)
f = sup : x N, x 0 … (i)
x

= sup f(x) : x 1

= inf k, k 0 and f(x) k x x

Proof: As we know that if N, N are normed linear spaces, (N, N ) is a normed linear space. If
N is a Banach space, (N, N ) is Banach space. Hence (N, R) or (N, C) is a Banach space because
R and C are Banach spaces even if N is not complete.
This completes the proof of the theorem.
Theorem 2: Let f be a linear functional on a normed linear space. If f is continuous at x N, it
o
must be continuous at every point of N.
Proof: If f is continuous at x = x , then
o
x x f (x ) f (x)
n o n
To show that f is continuous everywhere on N, we must show that for any y N,
y y f (y ) f (y)
n n
Let y y as n
n
Now f (y ) = f (y – y + x + y – x )
n n o o
since f is linear.
f (y ) = f (y – y + x ) + f (y) – f (x ) … (1)
n n o o
As y y y – y + x x by hypothesis
n n o o
Also f is continuous, f (y – y + x ) f (x ) … (2)
n o o
From (1) and (2), it follows that
f (y ) f (y) as n .
n
f is continuous at y N and consequently as it is continuous everywhere on N.
Hence proved.

16.1.2 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 .

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