Page 193 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes We first note that the above norm is well defined. Since f is bounded, we have
|f (x)| M || x||, M 0.

f(x)
Let M be the set of real numbers M satisfying this relation. Then the set ; x 0 is
x
bounded above so that it must possess a supremum. Let it be f . So f is well defined and we
must have

f(x)
f x 0.
x

or |f (x)| f x .

Let us check that defined by (1) is truly a norm on N*:

If f, g N*, then

f(x) g(x)
f g = sup
x 0 x

f(x) g(x)
sup sup
x 0 x x 0 x

f g f g .

Similarly, we can see that f f .

16.1.4 Equivalent Methods of Finding F

If f is a bounded linear functional on N, then
|f (x)| M x , M 0.

(I) f = inf {M : M M } where M is the set of all real numbers satisfying

|f (x)| M x ,

Since f M and M is the set of all non-negative real numbers, it is bounded below by
zero so that it has an infimum. Hence
f inf {M : M M } … (2)

f(x)
For x 0 and M M we have M. Since M is the only upper bound then from
x
definition (2), we have

f(x)
M sup = f for any M M .
x 0 x

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