Page 181 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 15: Banach Space: Definition and Some Examples

CONTENTS
Objectives
Introduction

15.1 Banach Spaces
15.1.1 Normed Linear Space
15.1.2 Convergent Sequence in Normed Linear Space

15.1.3 Subspace of a normed Linear Space
15.1.4 Complete Normed Linear Space
15.1.5 Banach Space
15.2 Summary
15.3 Keywords

15.4 Review Questions
15.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Know about Banach spaces.
 Define Banach spaces.
 Solve problems on Banach spaces.

Introduction

Banach space is a linear space, which is also, in a special way, a complete metric space. This
combination of algebraic and metric structures opens up the possibility of studying linear
transformations of one Banach space into another which have the additional property of being
continuous. The concept of a Banach space is a generalization of Hilbert space. A Banach space
assumes that there is a norm on the space relative to which the space is complete, but it is not
assumed that the norm is defined in terms of an inner product. There are many examples of
Banach spaces that are not Hilbert spaces, so that the generalization is quite useful.

15.1 Banach Spaces

15.1.1 Normed Linear Space

Definition: Let N be a complex (or real) linear space. A real valued function n : N R is said to
define, a norm on N if for any x, y N and any scalar (complex number) , the following
conditions are satisfied by n:

(i) n (x) 0, n (x) = 0, x = 0;
(ii) n (x + y) n (x) + n (y); and

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