Page 212 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 17: The Hahn-Banach Theorem

g (x + y) + i h (x + y) = g (x) + i h (x) + g (y) + i h (y) Notes
= g (x) + g (y) + i (h (x) + h (y))
Equating the real and imaginary parts, we get
g (x + y) = g (x) + g (y)

and h (x + y) = h (x) + h (y)
If R, then we have
f ( x) = g ( x) + i h ( x)
Since f is linear

f ( x) = f (x) = g (x) + i h (x)
f ( x) = g (x) and h ( x) = h (x)
(equating real and imaginary parts)

g, h are real linear functions on M.
Further |g (x)| |f (x)| f x
If f is bounded on M, then g is also bounded on M.
Similarly h is also bounded on M.

Since a complex linear space can be regarded as a real linear space by restricting the scalars to be
real numbers, we consider M as a real linear space. Hence g and h are real functional on real
space M.
For all x in M we have
f (i x) = i f (x) = i {g (x) + i h (x)}
or g (i x) + i h (i x) = – h (x) + i g (x)

Equating real and imaginary parts, we get
g (i x) = – h (x) and h (i x) = g (x)
Therefore we can express f (x) either only by g or only h as follows:
f (x) = g (x) – i g (i x)

= h (i x) + i h (x).
Since g is a real functional on M, by case I, we extend g to a real functional g on the real space M
o o
such that g = g . For x M , we define
o o
f (x) = g (x) – i g (i x)
o o o
First note that f is linear on the complex linear space M . Such that f = f on M.
o o o
Now f (x + y) = g (x + y) – i g (i x + i y)
o o o
= g (x) + g (y) – i g (i x) – i g (i y)
o o o o
= f (x) + f (y).
o o
Now for a, b R, we have
f ((a + i b) x) = g (ax + i bx) – i g (– bx + i ax)
o o o
= a g (x) + b g (i x) – i (–b) g (x) – i a g (i x)
o o o o
= (a + ib) {g (x) – i g (i x)}
o o

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