Page 199 - DMTH401_REAL ANALYSIS

Richa Nandra, Lovely Professional University Unit 15: Uniform Convergence of Functions

Unit 15: Uniform Convergence of Functions Notes

CONTENTS
Objectives
Introduction
15.1 Uniform Convergence
15.2 Testing Pointwise and Uniform Convergence
15.3 Covers and Subcovers

15.4 Dini’s Theorem
15.5 Summary
15.6 Keywords
15.7 Review Questions
15.8 Further Readings

Objectives

After studying this unit, you will be able to:
 Define uniform convergence

 Explain the testing pointwise and uniform convergence
 Discuss the covers and subcovers
 Explain Dini's theorem
Introduction

In earlier unit as you all studied about sequences in metric spaces. This unit will explain that
pointwise convergence of a sequence of functions was easy to define, but was too simplistic of a
concept. We would prefer a type of convergence that preserves at least some of the shared
properties of a function sequence. Such a concept is uniform convergence.

15.1 Uniform Convergence

Definition 1: Let I and { } ¥ be a sequence of real-valued functions on I. We say that f
f
n n = 1 n
converges to f pointwise on I as n ¥if:
" ÎI : f ( ) f ( ) as  ¥
x
x
x
n
x
f
x
Example: Let I = (0, 1) and f n ( ) = x n , f ( ) = x 2 , ( ) = x 3 ,.... ). It can be observed that
x
( 1
3
x
" ÎI : f n ( ) = x n  0. So f converges pointwise to the zero function on I.
x
n

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