Page 190 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Sachin Kaushal, Lovely Professional University Unit 16: Continuous Linear Transformations

Unit 16: Continuous Linear Transformations Notes

CONTENTS
Objectives
Introduction

16.1 Continuous Linear Transformation
16.1.1 Continuous Linear Functionals Definition
16.1.2 Bounded Linear Functional

16.1.3 Norm of a Bounded Linear Functional
16.1.4 Equivalent Methods of Finding F
16.1.5 Representation Theorems for Functionals
16.2 Summary
16.3 Keywords

16.4 Review Questions
16.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand continuous linear transformation
 Define bounded linear functional and norm of a bounded linear functional
 Understand theorems on continuous linear transformations.

Introduction

In this unit, we obtain the representation of continuous linear functionals on some of Banach
spaces.

16.1 Continuous Linear Transformation

16.1.1 Continuous Linear Functionals Definition

 Let N be a normed linear space. Then we know the set R of real numbers and the set C of
complex numbers are Banach spaces with the norm of any x R or x C given by the
absolute value of x. Thus with our previous notations, (N, R) or (N, C) denote respectively
the set of all continuous linear transformations from N into R or C.

 We denote the Banach space (N, R) or (N, C) by N* and call it by the conjugate space (or
dual space or adjoint space) of N.

 The elements of N* will be referred to as continuous linear functionals or simply functionals
on N.

LOVELY PROFESSIONAL UNIVERSITY 183