Page 323 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 30: Projections

CONTENTS
Objectives

Introduction
30.1 Projections
30.1.1 Perpendicular Projections

30.1.2 Invariance
30.1.3 Orthogonal Projections
30.2 Summary
30.3 Keywords

30.4 Review Questions
30.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Define perpendicular projections.
 Define invariance and orthogonal projections.
 Solve problems on projections.

Introduction

We have already defined projections both in Banach spaces and Hilbert spaces and explained
how Hilbert spaces have plenty of projection as a consequence of orthogonal decomposition
theorem or projection theorem. Now, the context of our present work is the Hilbert space H, and
not a general Banach space, and the structure which H enjoys in addition to being a Banach space
enables us to single out for special attention those projections whose range and null space are
orthogonal. Our first theorem gives a convenient characterisation of these projections.

30.1 Projections

30.1.1 Perpendicular Projections

A projection P on a Hilbert space H is said to be a perpendicular projection on H if the range M
and null space N of P are orthogonal.

Theorem 1: If P is a projection on a Hilbert space H with range M and null space N then M N P
is self-adjoint and in this case N M .

Proof: Let M N and z be any vector in H. Then since H M N, we can write z uniquely as

z x y,x M,y N.

316 LOVELY PROFESSIONAL UNIVERSITY