Page 211 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
x x
sup f f x o r o
y M

x x
inf f f x … (7)
y M o
If > 0 the right hand side of (7) becomes

1 1
r f(x) f x x o which implies that
o
f (x) + r = f (x + x ) f x + x
o o o o
If z = x + x M then we get from above
o o
|f (z)| f z … (8)
o
If < 0, then from L.H.S. of (7) we have

x x
f f x o r o

1 1 1 1
f(x) f x x o r , since < 0, .
o
f (x) + r f x + x
o o
f (z) f z for every z M … (9)
o o
Replacing z by –z in (9) we get

– f (z) f z , since f is linear on M … (10)
o o o
Hence we get from (9) and (10).
|f (z)| f z … (11)
o
Since f is functional on M, f is bounded.

Thus ( ) shows that f is a functional on M .
o o
Since f = sup {|f (z)| : z M , z |}, it follows from ( ) that
o o o
f f
o
We finally obtain from (A) and (B) that

f = f
o
This power the lemma for real normed linear space.
Case II: Let N be a complex normed linear space.
Let N be a normed linear space over C and f be a complex valued functional on a subspace M of
N.
Let g = Re (f) and h = Im (f) so that we can write
f (x) = g (x) + i h (x). We show that g (x) and h (x) are real valued functionals.

Since f is linear, we have
f (x + y) = f (x) + f (y)

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