Page 79 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 7: Convergence and Completeness

CONTENTS
Objectives
Introduction

7.1 Convergence and Completeness
7.1.1 Convergent Sequence
7.1.2 Cauchy Sequence

7.1.3 Complete Normed Linear Space
7.1.4 Banach Space
7.1.5 Summable Series
7.1.6 Riesz-Fischer Theorem
7.2 Summary

7.3 Keywords
7.4 Review Questions
7.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Understand convergence and completeness.
 Understand Riesz-Fischer theorem.
 Solve problems on convergence and completeness.

Introduction

Convergence of a sequence of functions can be defined in various ways, and there are situations
in which each of these definitions is natural and useful. In this unit, we shall start with the
p
definition of convergence and Cauchy sequence and proceed with the topic completeness of L .
7.1 Convergence and Completeness

7.1.1 Convergent Sequence

Definition: A sequence <x > in a normal linear space X with norm . is said to converge to an
n
element x X if for arbitrary > 0, however small, n N such that x – x < , n > n .
0 n 0
Then we write lim x x .
n n

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