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Richa Nandra, Lovely Professional University Unit 30: Ascoli’s Theorem

Unit 30: Ascoli’s Theorem Notes

CONTENTS
Objectives
Introduction

30.1 Ascoli’s Theorem
30.1.1 Equicontinuous
30.1.2 Uniformly Equicontinuous

30.1.3 Statement and Proof of Ascoli’s Theorem
30.2 Summary
30.3 Keywords
30.4 Review Questions
30.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Define equicontinuous and uniformly equicontinuous;
 Understand the proof of Ascoli’s theorem;
 Solve the problems on Ascoli’s theorem.

Introduction

Ascoli’s theorem deals with continuous functions and states that the space of bounded,
equicontinuous functions is compact. The space of bounded “equimeasurable functions,” is
compact and it contains the bounded equicontinuous functions as a subset. Giulio Ascoli is an
Italian Jewish mathematician. He introduced the notion of equicontinuity in 1884 to add to
closedness and boundedness for the equivalence of compactness of a function space. This is what
is called Ascoli’s theorem.

30.1 Ascoli’s Theorem

30.1.1 Equicontinuous

A family F of functions on a metric space (X, d) is called equicontinuous if

 x  X,   > 0,   > 0 s.t.  y  X with d(x, y) <  we have f(x) f(y)- <  for all fF.

30.1.2 Uniformly Equicontinuous

A family F of functions on a metric space (X, d) is called uniformly equicontinuous if   > 0,
  > 0 s.t.  x, y  X with d(x, y) < . We have f(x) f(y)- <  for all fF.

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