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Functional Analysis Notes (2011) Mr. Andrew Pinchuck
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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
1.2 Subsets of a linear space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Subspaces and Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Quotient Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Direct Sums and Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 The H¨older and Mi nkowski Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Normed Linear Spaces 13
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
3.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Best Approximation in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Orthonormal Sets and Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Bounded Linear Operators and Functionals 62
4.1 Intr oduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Examples of Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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
1
2011 FUNCT IONAL ANALYSIS ALP
6 Baire’s Category Theorem and it s Applications 99
6.1 Intr oduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 Uniform Boundedness Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3 The Open Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 Closed Graph Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2
2011 FUNCT IONAL ANALYSIS ALP
Introduction
These course notes are adapted from the original course notes written by Prof. Sizwe Mabizela when
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
differential equations, probability theory, quant um mechanics and measure theory amongst other areas and
can best be viewed as a powerful collection of too ls that have far reaching con sequences.
As a prerequ isite for this course, the reader must b e familiar with linear algebra up to t he level o f a
standard second year university course and be familiar with real analysis. The aim of this course is to
introduce the student to the key ideas of functional analy sis. It should be remembered however that we
only scratch the surface of this vast area in this course. We examine normed linear spaces, Hilbert sp aces,
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-
Toronto, 1978.
G. F. SIMMONS, Introduction to topology and modern analysis, McGraw-Hill Book Company, Sin-
gapore, 1963.
A. E. TAYLOR, Introduction to Functional Analysis, John Wiley & Sons, N. Y. 1958.
I have also found Wikipedia to be quite useful as a general ref erence.
1
Chapter 1
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.
Unless otherwise stated, we shall denote by R the field of real numbers and by C the field of complex
numbers. Let F denote either R or C.
1.1.1 Definition
A
linear space
over a field
F
is a nonempty set
X
with two operations
C W X X ! X
(called addition)
;
and
W F X ! X
(called multiplication)
satisfying the fol lowing properties:
[1]
x C y 2 X
whenever
x; y 2 X
;
[2]
x C y D y C x
for all
x; y 2 X
;
[3] There exists a unique element in
X
, denoted by 0, such that
x C 0 D 0 C x D x
for al l
x 2 X
;
[4] Associated with each
x 2 X
is a unique element in
X
, denoted by
x
, such t hat
x C .x/ D
x C x D 0
;
[5]
.x C y/ C z D x C .y C z/
for all
x; y; z 2 X
;
[6]
˛ x 2 X
for all
x 2 X
and for all
˛ 2 F
;
[7]
˛ .x C y/ D ˛ x C ˛ y
for all
x; y 2 X
and all
˛ 2 F
;
[8]
.˛ C ˇ/ x D ˛ x C ˇ x
for all
x 2 X
and all
˛; ˇ 2 F
;
[9]
.˛ˇ/ x D ˛ .ˇ x/
for all
x 2 X
and all
˛; ˇ 2 F
;
[10]
1 x D x
for all
x 2 X
.
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.
2
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