Page 213 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes So that
f ((a + i b) x) = (a + i b) f (x)
o o
f is linear on M and also g = g on M.
o o o
f = f on M.
o
Now have to show that f = f on M .
o o
Let x M and f (x) = re i
o o
|f (x) | = r = e re = e f (x) … (12)
i
–i
–i
o o
Since f (x) is linear,
o
e f (x) = f (e x) … (13)
–i
–i
o o
So we get from (12) and (13) that
–i
|f (x) | = r = f (r e x).
o o
Thus the complex valued functional f is real and so it has only real part so that
o
|f (x) | = g (e x) |g (e x)|
–i
–i
o o o
–i
–i
But |g (e x)| g e x
o o
= g x ,
o
We get |f (x)| g x
o o
Since g is the extension of g, we get
o
g x = g x f x
o
Therefore
|f (x) f x so that from the definition of the norm of f , we have
o o
f f
o
As in case I, it is obvious that f f
o
Hence f = f .
o
This completes the proof of the theorem.
17.1.2 Theorems and Solved Examples
Theorem: The generalized Hahn-Banach Theorem for Complex Linear Space.
Let L be a complex linear space. Let p be a real valued function defined on L such that
p (x + y) p (x) + p (y)
and p ( x) = | | p (x) x L and scalar .

If f is a complex linear functional defined on the subspace M such that |f (x)| p (x) for x M,
then f can be extended to a complex linear functional to be defined on L such that |f (x)| p (x)
o
for every x L.
Proof: We have from the given hypothesis that f is a complex linear functional on M such that

| f (x) p (x) x M.

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