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Richa Nandra, Lovely Professional University Unit 17: The Hahn-Banach Theorem

Unit 17: The Hahn-Banach Theorem Notes

CONTENTS
Objectives
Introduction

17.1 The Hahn-Banach Theorem
17.1.1 Theorem: The Hahn-Banach Theorem – Proof
17.1.2 Theorems and Solved Examples

17.2 Summary
17.3 Keywords
17.4 Review Questions
17.5 Further Readings

Objectives

After studying this unit, you will be able to:

 State the Hahn-Banach theorem
 Understand the proof of the Hahn-Banach theorem

 Solve problems related to it.
Introduction

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. Although there are many forms of Hahn-Banach theorem, however we are interested in
Banach space theory, in which we shall first prove Hahn-Banach theorem for normed linear
spaces and then prove the generalised form of this theorem. In the next unit, we shall discuss
some important applications of this theorem.

17.1 The Hahn-Banach Theorem

17.1.1 Theorem: The Hahn-Banach Theorem – Proof

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 on the whole space N such that
o
f = f .
o
Proof: We first prove the following lemma which constitutes the most difficult part of this
theorem.

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