Page 232 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Richa Nandra, Lovely Professional University Unit 20: The Closed Graph Theorem

Unit 20: The Closed Graph Theorem Notes

CONTENTS
Objectives
Introduction

20.1 The Closed Graph Theorem
20.1.1 Graph of Linear Transformation
20.1.2 Closed Linear Transformation

20.1.3 The Closed Graph Theorem - Proof
20.2 Summary
20.3 Keyword
20.4 Review Questions
20.5 Further Readings

Objectives

After studying this unit, you will be able to:
 State the closed graph theorem.
 Understand the proof of the closed graph theorem
 Solve problems based on the closed graph theorem.

Introduction

Though many of the linear transformations in analysis are continuous and consequently bounded,
there do exist linear transformation which are discontinuous. The study of such kind of
transformation is much facilitated by studying the graph of transformation and using the graph
of the transformation as subset in the Cartesian product space to characterise the boundedness of
such transformations. The basic theorem in this regard is the closed graph theorem.

20.1 The Closed Graph Theorem

20.1.1 Graph of Linear Transformation

Definition: Let N and N be a normed linear space and let T : N N be a mapping with domain
N and range N . The graph of T is defined to be a subset of N × N which consists of all ordered
pairs (x, T (x)). It is generally denoted by G .
T
Therefore the graph of T : N N is

G = {(x, T (x) : x N}.
T

Notes G is a linear subspace of the Cartesian product N × N with respect to coordinate-
T
wise addition and scalar multiplications.

LOVELY PROFESSIONAL UNIVERSITY 225