Functional Analysis Notes (2011) Mr. Andrew Pinchuck

FUNCTIONAL ANALYSIS NOTES

(2011)

Mr. Andrew Pinchuck

Department of Mathematics (Pure & Applied)

Rhodes U n iversity

Contents

Introducti on 1

1 Linear Spaces 2

1.1 Intr oducton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Quotient Norm and Quotient Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Completeness of Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Series in Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Boun ded, Totally Bounded, and Compact Subsets of a Normed Linear Space . . . . . . . 26

2.6 Finite Dimensional N ormed Li near Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Separable Spaces and Schauder Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Hilbert Spaces 36

3.1 Intr oduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Completeness of Inner Prod uct Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 The Dual Space of a Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 The Hahn-Banach Theorem and its Consequences 81

5.1 Intr oduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Consequences of the Hahn-Banach Extension Theorem . . . . . . . . . . . . . . . . . . . 85

5.3 Bidual of a normed linear space and Reflexivity . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 The Adj oint Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.5 Weak Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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2011 FUNCT IONAL ANALYSIS ALP

6 Baire’s Category Theorem and it s Applications 99

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he last gave this course in 2006 to whom I am indebted. I thus make no claims of originality but have made

several changes throughout. In particular, I have attempted to motivate these results in terms of applications

in science and in other important branches of mathematics.

Functional analysis is the branch of mathematics, specifically of analysis, concerned with the study of

vector spaces and operators acting on them. It is essentially where linear algebra meets analysis. That is,

an important part of fu nctional analysis is the study of vector spaces endowed with topological structure.

Functional analysis arose in the study of tansformations of functions, such as the Fourier transform, and in

the study of differential and integral equations. The founding and early development of functi onal analysis

is l argely du e to a group o f Polish mathematicians around Stefan Banach in the first half of the 20th century

but continues to be an area of intensive research to this day. Functional analysis has its main applications in

bounded linear operators, dual spaces and the most famous and important results in functional analysis

such as the Hahn-Banach theorem, Baires category theorem, the uniform boundedness principle, the open

mapping theorem and the closed graph theorem. We attempt to give justifications and motivations for the

ideas developed as we go along.

Throughout the notes, you will notice that there are exercises and it is up to the stud ent to work through

these. In certain cases, there are statements made with out justification and once again it is up to the stud ent

to rigourously verify these results. For f urther reading on th ese topics the reader is referred to the following

texts:

 G. BACHMAN, L. N ARICI, Functional Analysis, Academic Press, N.Y. 1966.

 E. KREYSZIG, Introductory Functional Analysis, John Wiley & sons, New York-Chichester-B risbane-

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Linear Spaces

1.1 Introducton

In this first chapter we r eview the important notions associated with vector spaces. We also state and prove

some well known inequalities that will have importan t consequences in the following chapter.

X

C W X  X ! X

;

 W F  X ! X

x C 0 D 0 C x D x

x 2 X

x 2 X

X

x

x; y; z 2 X

˛ 2 F

x; y 2 X

˛ 2 F

We emphasize that a linear space is a quadruple .X; F; C; / where X is the underly ing set, F a field, C

addition, and  mult iplication. When no confusion can arise we shall identify the linear space .X; F; C; /

with the underlying set X . To show that X is a linear space, it suffices to show that it is closed under

addition and scalar multiplication operations. Once thi s has been shown, it is easy to show that all the other

axioms hold.

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