Page 105 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
x (1/n) x (1/n)
Also |G (x)| = f(t) dt n |f(t)|dt M .
n
x x
Hence |G (x)| M, n N and x [a, b].
n
2
2
[G (x) – f (x)] 2 (M + M) = 4M , x [a, b].
n
On applying Lebesgue bounded convergence theorem, we get
b b
Lim (G n f) 2 = Lim (G n f) 2 0
n
a a
2
Lim G n f = 0
n 2
Lim G n f
n 2
or Lim f G n = 0
n
for given > 0, n N, such that n n
o o
f – G <
n 2
Particularly for n = n .
o
f G <
n o 2
f – g < (Taking G = g)
2 n o
Thus there exists a continuous function G (x) g(x)
n o
x (1/n )
o
= n o f(t) dt, x [a, b],
x
which satisfies the given condition.

8.2 Summary

 A linear functional f on a normed space N is said to be bounded if there is a constant
1
k > 0 such that
|f (x)| k x , x N
1
 If x  and f is bounded linear functional on  , then f has the unique representation of
p p
the form as an infinite series.

f (x) = x f(e )
k
k
k 1
*
 The norm of f  is given by
p
1
q
f = |f(e )| q
k
k 1

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