Page 245 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 22: The Uniform Boundedness Theorem

CONTENTS
Objectives
Introduction
22.1 The Uniform Boundedness Theorem
22.1.1 The Uniform Boundedness Theorem – Proof

22.1.2 Theorems and Solved Examples
22.2 Summary
22.3 Keywords
22.4 Review Questions
22.5 Further Readings

Objectives

After studying this unit, you will be able to:

 State the uniform boundedness theorem.
 Understand the proof of this theorem.
 Solve problems related to uniform boundedness theorem.

Introduction

The uniform boundedness theorem, like the open mapping theorem and the closed graph
theorem, is one of the cornerstones of functional analysis with many applications. The open
–1
mapping theorem and the closed graph theorem lead to the boundedness of T whereas the
uniform boundedness operators deduced from the point-wise boundedness of such operators.
In uniform boundedness theorem we require completeness only for the domain of the definition
of the bounded linear operators.

22.1 The Uniform Boundedness Theorem

22.1.1 The Uniform Boundedness Theorem – Proof

If (a) B is a Banach space and N a normed linear space,
(b) {T } is non-empty set of continuous linear transformation of B into N, and
i
(c) {Ti (x)} is a bounded subset of N for each x B, then { T } is a bounded set of numbers, i.e.
i
{T } is bounded as a subset of (B, N)
i
Proof: For each positive integer n, let
F = {x B : T (x) n i}.
n i
Then F is a closed subset of B. For if y is any limit point of F , then a sequence (x ) of points of
n n k
F such that
n
x y as k
k

238 LOVELY PROFESSIONAL UNIVERSITY