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Richa Nandra, Lovely Professional University Unit 18: The Natural Imbedding of N in N**

Unit 18: The Natural Imbedding of N in N** Notes

CONTENTS
Objectives
Introduction

18.1 The Natural Imbedding of N into N**
18.1.1 Definition: Natural Imbedding of N into N**
18.1.2 Definition: Reflexive Mapping

18.1.3 Properties of Natural Imbedding of N into N**
18.1.4 Theorems and Solved Examples
18.2 Summary
18.3 Keywords
18.4 Review Questions

18.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define the natural imbedding of N into N**.
 Define reflexive mapping.
 Describe the properties of natural imbedding of N into N*.

Introduction

As we know that conjugate space N* of a normed linear space N is itself a normed linear space.
So, we can find the conjugate space (N*)* of N*. We denote it by N** and call it the second
conjugate space of N. Likewise N*, N** is also a Banach space. The importance of the space N**
lies in the fact that each vector x in N given rise to a functional F in N** and that there exists an
x
isometric isomorphism of N into N**, called natural imbedding of N into N**.
18.1 The Natural Imbedding of N into N**

18.1.1 Definition: Natural Imbedding of N into N**

The map J : N N** defined by

J (x) = F x N,
x
is called the natural imbedding of N into N**.
Since J (N) N**, N can be considered as part of N** without changing its basic norm structure.
We write N N** in the above sense.

LOVELY PROFESSIONAL UNIVERSITY 211