Page 229 - DMTH401_REAL ANALYSIS

Sachin Kaushal, Lovely Professional University Unit 18: Equicontinuous

Unit 18: Equicontinuous Notes

CONTENTS
Objectives
Introduction
18.1 Equicontinuity
18.2 Families of Equicontinuous
18.3 Equicontinuity and Uniform Convergence

18.4 Equicontinuity Families of Linear Operators
18.5 Equicontinuity in Topological Spaces
18.6 Stochastic Equicontinuity
18.7 Summary
18.8 Keywords
18.9 Review Questions
18.10 Further Readings

Objectives

After studying this unit, you will be able to:

 Explain the equicontinuity
 Describe the properties of equicontinuous
 Discuss the equicontinuity and uniform convergence
 Define stochastic equicontinuity

Introduction

In last unit, you have studied about the uniform converges and differentiation. This unit provides
you the explanation of Equicontinuity. In mathematical analysis, a family of functions is
equicontinuous if all the functions are continuous and they have equal variation over a given
neighbourhood, in a precise sense described herein. In particular, the concept applies to countable
families, and thus sequences of functions.

18.1 Equicontinuity

The equicontinuity appears in the formulation of Ascoli’s theorem, which states that a subset of
C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only
if it is closed, pointwise bounded and equicontinuous. A sequence in C(X) is uniformly convergent
if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous
a-prior). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous
functions f on either metric space or locally compact space is continuous. If, in addition, f are
n n
homomorphic, then the limit is also homomorphic.
The uniform boundedness principle states that a pointwise bounded family of continuous linear
operators between Banach spaces is equicontinuous.

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