Page 269 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Richa Nandra, Lovely Professional University

Notes Unit 25: Orthonormal Sets

CONTENTS
Objectives
Introduction

25.1 Orthonormal Sets
25.1.1 Unit Vector or Normal Vector
25.1.2 Orthonormal Sets, Definition

25.1.3 Examples of Orthonormal Sets
25.1.4 Theorems on Orthonormal Sets
25.2 Summary
25.3 Keywords
25.4 Review Questions

25.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand orthonormal sets
 Define unit vector or normal vector
 Understand the theorems on orthonormal sets.

Introduction

In linear algebra two vectors in an inner product space are orthonormal if they are orthogonal
and both of unit length. A set of vectors from an orthonormal set if all vectors in the set are
mutually orthogonal and all of unit length.
In this unit, we shall study about orthonormal sets and its examples.

25.1 Orthonormal Sets

25.1.1 Unit Vector or Normal Vector

Definition: Let H be a Hilbert space. If x H is such that x = 1, i.e. (x, x) = 1, then x is said to be
a unit vector or normal vector.

25.1.2 Orthonormal Sets, Definition

A non-empty subset { e } of a Hilbert space H is said to be an orthonormal set if
i
(i) i j e e, equivalently i j (e , e) = 0
i j i j
(ii) e = 1 or (e , e) = 1 for every i.
i i j

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