Page 216 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 17: The Hahn-Banach Theorem

Hence by Hahn-Banach theorem, f can be extended to a functional f N* such that f (M) = f (M) Notes
o o
and f = f = 1, which in particular yields that
o
f (x ) = f (x ) = x and f = 1.
o o o o o
This completes the proof of the theorem.
Corollary 2: N* separates the vector (points) in N.
Proof: To prove the cor. it suffices to show that if x, y N with x y, then there exists a f N*
such that f (x) f (y).
Since x y x – y 0.
So by above theorem there exists a functional f N* such that
f (x – y) = f (x) – f (y) 0

and hence f (x) f (y).
This shows that N* separates the point of N.
Corollary 3: If all functional vanish on a given vector, then the vector must be zero, i.e.

if f (x) = 0 f N* then x = 0.
Proof: Let x be the given vector such that f (x) = 0 f N*.
Suppose x 0. Then by above theorem, there exists a function f N* such that

f (x) = x > 0, which contradicts our supposition that
f (x) = 0 f N*. Hence we must have x = 0.

17.2 Summary

 The Hahn-Banach Theorem: Let N be a normed linear space and M be a linear subspace of
N. If f is a linear functional defined on M, then f can be extended to a functional f defined
o
on the whole space N such that
f = f
o
 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 function f defined on L such that
o
| f (x) | p (x) for every x L.
o
17.3 Keywords

Hahn-Banach theorem: The Hahn-Banach theorem is one of the most fundamental and important
theorems in functional analysis. It is most fundamental in the sense that it asserts the existence
of the linear, continuous and norm preserving extension of a functional defined on a linear
subspace of a normed linear space and guarantees the existence of non-trivial continuous linear
functionals on normed linear spaces.
Sub-linear Functional on L: Let L be a linear space. A mapping p : L R is called a sub-linear
functional on L if it satisfies the following two properties namely,
(i) p (x + y) p (x) + p (y) x, y L (sub additivity)

(ii) p ( x) = p (x), 0 (positive homogeneity)

LOVELY PROFESSIONAL UNIVERSITY 209