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Richa Nandra, Lovely Professional University Unit 26: The Conjugate Space H*

Unit 26: The Conjugate Space H * Notes

CONTENTS

Objectives
Introduction
26.1 The Conjugate Space H *

26.1.1 Definition
26.1.2 Theorems and Solved Examples
26.2 Summary

26.3 Keywords
26.4 Review Questions
26.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Define the conjugate space H . *

 Understand theorems on it.
 Solve problems related to conjugate space H . *

Introduction

Let H be a Hilbert space. A continuous linear transformation from H into C is called a continuous
linear functional or more briefly a functional on H. Thus if we say that f is a functional on H, then
f will be continuous linear functional on H. The set H,C of all continuous linear functional on
H is denoted by H and is called the conjugate space of H. The elements of H are called continuous
*
*
linear functional or more briefly functionals. We shall see that the conjugate space of a Hilbert
*
space H is the conjugate space H of H is in some sense is same as H itself. After establishing a
correspondence between H and H , we shall establish the Riesz representation theorem for
*
*
continuous linear functionals. Thereafter we shall prove that H is itself a Hilbert space and H is
reflexive, i.e. has a natural correspondence between H and H and this natural correspondence
**
is an isometric isomorphism of H onto H .
**
26.1 The Conjugate Space H *

26.1.1 Definition

Let H be a Hilbert space. If f is a functional on H, then f will be continuous linear functional on
H. The set H,C of all continuous linear functional on H is denoted by H and is called the
*
conjugate space of H. The conjugate space of a Hilbert space H is the conjugate space H of H is in
*
some sense is same as H itself.

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