Page 217 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Thus p defined on L in the above theorems is a sub-linear functional on L.
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 .

17.4 Review Questions

1. Let M be a closed linear subspace of a normed linear space N and x is a vector not in M.
o
Then there exists a functional f N* such that
o
f (M) = 0 and f (x ) 0
o o o
2. Let M be a closed linear subspace of a normed linear space N, and let x be a vector not in
o
M. If d is the distance from x to M, then these exists a functional f N* such that
o o
f (M) = 0, f (x ) = d, and f = 1.
o o o o
3. Let M is a closed linear subspace of a normed linear space N and x N such that x M.
o o
If d is the distance from x to M, then there exists a functional f N* such that f (M) = 0, f
o o o o
1
(x ) = 1 and f = .
o o d
4. Let N be a normed linear space over R or C. Let M N be a linear subspace. Then
M = N f N* is such that f (x) = 0 for every x M, then f = 0.

5. A normed linear space is separable if its conjugate (or dual) space is separable.
17.5 Further Readings

Books Walter Rudin, Functional Analysis (2nd ed.). McGraw-Hill Science/Engineering/
Math 1991.
Eberhard Zeidler, Applied Functional Analysis: Main Principles and their Applications,
Springer, 1995.

Online links mat.iitm.ac.in
www.math.ksu.edu
mizar.uwb.edu.pl/JFW/Vol5/hahnban.html

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