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Richa Nandra, Lovely Professional University Unit 19: The Open Mapping Theorem

Unit 19: The Open Mapping Theorem Notes

CONTENTS
Objectives
Introduction

19.1 The Open Mapping Theorem
19.1.1 Lemma
19.1.2 Proof of the Open Mapping Theorem

19.1.3 Theorems and Solved Examples
19.2 Summary
19.3 Keywords
19.4 Review Questions
19.5 Further Readings

Objectives

After studying this unit, you will be able to:
 State the open mapping theorem.

 Understand the proof of the open mapping theorem.
 Solve problems on the open mapping theorem.
Introduction

In this unit, we establish the open mapping theorem. It is concerned with complete normed
linear spaces. This theorem states that if T is a continuous linear transformation of a Banach
space B onto a Banach space B , then T is an open mapping. Before proving it, we shall prove a
lemma which is the key to this theorem.

19.1 The Open Mapping Theorem

19.1.1 Lemma

Lemma 1: If B and B are Banach spaces and T is a continuous linear transformation of B onto B ,
then the image of each sphere centered on the origin in B contains an open sphere centered on
the origin in B .
Proof: Let S and S respectively denote the open sphere with radius r centered on the origin in B
r r
and B .
We one to show that T (S ) contains same S .
r r
However, since T (S ) = T (r S ) = r T (S ), (by linearity of T).
r 1 1
It therefore suffices to show that T (S ) contains some S for then S , where = r , will be contained
2
1 r
in T (S ). We first claim that T(S ) (the closure of T (S )) contains some S .
r 1 1 r

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