Page 235 - DMTH401_REAL ANALYSIS

Sachin Kaushal, Lovely Professional University Unit 19: Arzelà’s Theorem and Weierstrass Approximation Theorem

Unit 19: Arzelà’s Theorem and Weierstrass Notes
Approximation Theorem

CONTENTS
Objectives

Introduction
19.1 Arzelà–Ascoli Theorem
19.2 Fourier Series
19.3 Summary
19.4 Keyword
19.5 Review Questions
19.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Discuss the Arzelà’s Theorem
 Describe the Weierstrass Approximation Theorem
Introduction

In last unit you have studied about the uniform convergence and Equicontinuity. This unit
provides you the explanation of Arzelà’s Theorem and Weierstrass Approximation theorem.
Our setting is a compact metric space X which you can, if you wish, take to be a compact subset
of Rn, or even of the complex plane (with the Euclidean metric, of course). Let C(X) denotes the
space of all continuous functions on X with values in C (equally well, you can take the values to
lie in R). In C(X) we always regard the distance between functions f and g in C(X) to be
dist(f, g) max{ f(x) g(x) : x X}
-
Î
=
19.1 Arzelà–Ascoli Theorem

A sequence {f } of continuous functions on an interval I = [a, b] is uniformly bounded if there
n n ÎN
is a number M such that
|f (x)|  M
n
for every function f belonging to the sequence, and every x Î [a, b]. The sequence is equicontinuous
n
if, for every  > 0, there exists a  > 0 such that
|f (x) – f (y)| <  Whenever |x – y| < 
n n
for every f belonging to the sequence. Succinctly, a sequence is equicontinuous if and only if all
n
of its elements have the same modulus of continuity. In simplest terms, the theorem can be
stated as follows:
Consider a sequence of real-valued continuous functions (f ) defined on a closed and bounded
n nÎN
interval [a, b] of the real line. If this sequence is uniformly bounded and equicontinuous, then
there exists a subsequence (f ) that converges uniformly.
nk

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