Page 57 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 5: Spaces, Hölder

CONTENTS
Objectives
Introduction

5.1 Spaces, Hölder
5.1.1 L -spaces
p
5.1.2 Conjugate Numbers

p
5.1.3 Norm of an Element of L -space
5.1.4 Simple Version of Hölder's Inequality
5.1.5 Hölder's Inequality
5.1.6 Riesz-Hölder's Inequality
5.1.7 Riesz-Hölder's Inequality for 0 < p < 1

5.2 Summary
5.3 Keywords
5.4 Review Questions

5.5 Further Readings
Objectives

After studying this unit, you will be able to:
p
p
 Understand L -spaces, conjugate numbers and norm of an element of L -space
 Understand the proof of Hölder’s inequality.
Introduction

In this unit, we discuss an important construction, which is extremely useful in virtually all
p
branches of analysis. We shall study about L -spaces and Hölder’s inequality.
5.1 Spaces, Hölder

5.1.1 L -Spaces
P
p
The class of all measurable functions f (x) is known as L -spaces over [a, b], if Lebesgue –
integrable over [a, b] for each p exists, 0 < p < , i.e.

b
p
|f| dx , (p 0)
a
and is denoted by L [a, b].
p

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