Page 194 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 16: Continuous Linear Transformations

Notes
Since M is bounded below by f , it has an infimum so that we have
inf M = inf {M : M M } f … (3)
M M
From (2) and (3), it follows that

f = inf {M : M M }

(II) f = sup f (x)
x 0

Let us consider x 1. Then

f(x) f x f .

Therefore, we have

sup f(x) f . … (4)
x
Now by definition,
f(x)
f = sup
x 0 x

It follows from the property of the supremum that, given > 0, an x N such that

f(x )
> ( f ) … (5)
x
Define

x
x . Then x is a unit vector.
x

Since x 1 x 1 , we have

1
sup f(x) f(x) f(x ) ( f ) [by (2)]
x 1 x

Hence > 0 is arbitrary, we have

sup f(x) > f … (6)
x 1
From (4) and (6), we obtain

sup f(x) = f .
x 1
(III) f = sup f(x) .
x 1
Consider x = 1, we have

f (x) f x f

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