Page 89 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Richa Nandra, Lovely Professional University

Notes Unit 8: Bounded Linear Functional on the L -spaces
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CONTENTS
Objectives
Introduction
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8.1 Bounded Linear Functionals on L -spaces
8.1.1 Linear Functional

8.1.2 Bounded Linear Functional
8.1.3 Bounded Linear Functional on L -spaces
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8.1.4 Norm
8.1.5 Continuous Linear Functional
8.1.6 Theorems
8.2 Summary
8.3 Keywords

8.4 Review Questions
8.5 Further Readings
Objectives

After studying this unit, you will be able to:
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 Understand bounded linear functional on L -spaces
 Understand related theorems.
 Solve problems on bounded linear functionals.

Introduction

In this unit, we obtain the representation of bounded linear functionals on L -space. We shall
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also study about linear functional, continuous linear functionals and norm of f  . Further, we
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shall prove important theorems on bounded linear functionals.
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8.1 Bounded Linear Functionals on L -spaces

8.1.1 Linear Functional

Definition: Let N be a normed space over a field R (or C). A mapping f : N R (or C) is called a
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linear functional on N if f ( x + y) = f (x) + f (y), x, y N and , R (or C).
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8.1.2 Bounded Linear Functional

Definition: A linear functional f on a normed space N is said to be bounded if there is a constant
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k > 0 such that
|f (x)| k x , x N … (1)
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