Page 251 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 23: Hilbert Spaces: The Definition and
Some Simple Properties

CONTENTS
Objectives

Introduction
23.1 Hilbert Spaces
23.1.1 Inner Product Spaces
23.1.2 Hilbert Space and its Basic Properties

23.1.3 Hilbert Space: Definition
23.1.4 Examples of Hilbert Space
23.2 Summary
23.3 Keywords

23.4 Review Questions
23.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define inner product spaces.
 Define Hilbert space.
 Understand basic properties of Hilbert space.
 Solve problems on Hilbert space.

Introduction

Since an inner product is used to define a norm on a vector space, the inner product are special
normed linear spaces. A complete inner product space is called a Hilbert space. We shall also see
from the formal definition that a Hilbert space is a special type of Banach space, one which
possesses additional structure enabling us to tell when two vectors are orthogonal. From the
above information, one can conclude that every Hilbert space is a Banach space but not conversely
in general.
We shall first define Inner Product spaces and give some examples so as to understand the
concept of Hilbert spaces more conveniently.

23.1 Hilbert Spaces

23.1.1 Inner Product Spaces

Definition: Let X be a linear space over the field of complex numbers C. An inner product on X is
a mapping from X × X C which satisfies the following conditions:

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