Functional analysis

127 Pages·2004·0.694 MB·English

by  Belton A.

Functional Analysis
Alexander C. R. Belton
Copyright c Alexander C. R. Belton 2004, 2006
(cid:13)
Hyperlinked and revised edition
All rights reserved
The right of Alexander Belton to be identified as the author
of this work has been asserted by him in accordance with
the Copyright, Designs and Patents Act 1988.
Contents
Contents i
Introduction iii
1 Normed Spaces 3
1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Subspaces and Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Initial Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Exercises 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Linear Operators 17
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Completeness of B(X,Y) . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Extension of Linear Operators . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 The Baire Category Theorem . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 The Open-Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Exercises 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 The Closed-Graph Theorem . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 The Principle of Uniform Boundedness . . . . . . . . . . . . . . . . . . 27
2.9 The Strong Operator Topology . . . . . . . . . . . . . . . . . . . . . . 27
2.10 Exercises 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Dual Spaces 31
3.1 Initial Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 The Weak Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Zorn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 The Hahn-Banach Theorem . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 The Weak Operator Topology . . . . . . . . . . . . . . . . . . . . . . . 36
3.6 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.7 The Weak* Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.8 Exercises 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.9 Tychonov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
i
ii Contents
3.10 The Banach-Alaoglu Theorem . . . . . . . . . . . . . . . . . . . . . . . 40
3.11 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.12 The Krein-Milman Theorem . . . . . . . . . . . . . . . . . . . . . . . . 45
3.13 Exercises 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Normed Algebras 53
4.1 Quotient Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Unitization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Approximate Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 Invertibility 59
5.1 The Spectrum and Resolvent . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 The Gelfand-Mazur Theorem . . . . . . . . . . . . . . . . . . . . . . . 61
5.3 The Spectral-Radius Formula . . . . . . . . . . . . . . . . . . . . . . . 61
5.4 Exercises 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 Characters and Maximal Ideals 65
6.1 Characters and the Spectrum . . . . . . . . . . . . . . . . . . . . . . . 66
6.2 The Gelfand Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.3 The Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . 68
6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.5 Exercises 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
via
A Tychonov Nets 75
A.1 Exercises A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Solutions to Exercises 79
Exercises 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Exercises 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Exercises 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Exercises 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Exercises 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Exercises 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Exercises 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Exercises A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Bibliography 117
Index 119
Introduction
These notes are an expanded version of a set written for a course given to final-year
undergraduates at the University of Oxford.
A thorough understanding of the Oxford third-year b4 analysis course (an introduction
to Banach and Hilbert spaces) or its equivalent is a prerequisite for this material. We
use [24] as a compendium of results from that series of lectures. (Numbers in square
brackets refer to items in the bibliography.)
The author acknowledges his debt to all those from whom he has learnt functional
analysis, especially Professor D. A. Edwards, Dr G. R. Allen and Dr J. M. Lindsay. The
students attending the course were very helpful, especially Mr A. Evseev, Mr L. Taitz
and Ms P. Iley.
This document was typeset using LATEX2ε with Peter Wilson’s memoir class and the
-LATEXandXY-picpackages. The index was produced withthe aidof theMakeIndex
AMS
program.
Alexander C. R. Belton
Lady Margaret Hall
Oxford
20th August 2004
This edition contains a few additional exercises and the electronic version is equipped
with hyperlinks, thanks to the hyperref package of Sebastian Rahtz and Heiko Oberdiek.
ACRB
University College
Cork
30th September 2006
Convention
ThroughoutthesenoteswefollowtheDirac-formalismconventionthatinnerproducts
on complex vector spaces are conjugate linear in the first argument and linear in the
second, in contrast to many Oxford courses.
iii

Spaces
1

One Normed Spaces
Throughout, the scalar field of a vector space will be denoted by F and will be either
the real numbers R or the complex numbers C.
Basic Definitions
Definition 1.1. A norm on a vector space X is a function
: X R+ := [0, ); x x
k·k → ∞ 7→ k k
that satisfies, for all x, y X and α F,
∈ ∈
(i) x = 0 if and only if x = 0 (faithfulness),
k k
(ii) αx = α x (homogeneity)
k k | |k k
and (iii) x+y 6 x + y (subadditivity).
k k k k k k
A seminorm on X is a function p: X R+ that satisfies (ii) and (iii) above.
→
Definition 1.2. Anormed vector space is a vector space X with a norm ; if necessary
k·k
wewilldenotethenormonthespaceX by . Wewillsometimes usethetermnormed
X
k·k
space as an abbreviation.
Definition 1.3. A Banach space is a normed vector space (E, ) that is complete,
k ·k
i.e., every Cauchy sequence in E is convergent, where E is equipped with the metric
d(x,y) := x y .
k − k
Definition 1.4. Let (x ) be a sequence in the normed vector space X. The series
n n>1
∞ x is convergent if there exists x X such that n x is convergent to x,
n=1 n ∈ k=1 k n>1
and the series is said to have sum x. The series is absolutely convergent if ∞ x is
P (cid:0)P (cid:1) n=1k nk
convergent.
P
Theorem 1.5. (Banach’s Criterion) A normed vector space X is complete if and
only if every absolutely convergent series in X is convergent.
Proof
This is a b4 result: see [24, Theorem 1.2.9]. (cid:3)
3
4 Normed Spaces
Subspaces and Quotient Spaces
Definition 1.6. A subspace of a vector space X is a subset M X that is closed under
⊆
vector addition and scalar multiplication: M + M M and αM M for all α F,
⊆ ⊆ ∈
where
A+B := a+b : a A, b B and αA := αa : a A A,B X, α F.
{ ∈ ∈ } { ∈ } ∀ ⊆ ∈
Example 1.7. Let (X, T) be a topological space and let (E, ) be a Banach space
E
k·k
over F. The set of continuous, E-valued functions on X forms an vector space, denoted
by C(X,E), where the algebraic operations are defined pointwise: if f, g C(X,E) and
∈
α F then
∈
(f +g)(x) := f(x)+g(x) and (αf)(x) := αf(x) x X.
∀ ∈
The subspace of bounded functions
C (X,E) := f C(X,E) f < ,
b ∞
∈ k k ∞
(cid:8) (cid:12) (cid:9)
where (cid:12)
: C (X,E) R+; f sup f(x) : x X ,
∞ b E
k·k → 7→ k k ∈
is a Banach space, with supremum norm (s(cid:8)ee Theorem 1.36)(cid:9). If X is compact
∞
k · k
then every continuous, E-valued function is bounded, hence C(X,E) = C (X,E) in this
b
case. If E = C (the most common case of interest) we use the abbreviations C(X) and
C (X).
b
Proposition 1.8. A subspace of a Banach space is closed if and only if it is complete.
Proof
See [24, Theorem 1.2.10]. (cid:3)
Definition 1.9. Given a vector space X with a subspace M, the quotient space X/M
is the set
X/M := [x] := x+M x X , where x+M := x+m : m M ,
∈ { ∈ }
(cid:8) (cid:12) (cid:9)
equipped with the vector-spac(cid:12)e operations
[x]+[y] := [x+y] and α[x] := [αx] x,y X, α F.
∀ ∈ ∈
(It is a standard result of linear algebra that this defines a vector space; for a full
discussion see [7, Appendix A.4].) The dimension of X/M is the codimension of M (in
X).
Theorem 1.10. Let X be a normed vector space with a subspace M and let
[x] := inf x m : m M [x] X/M.
X/M k − k ∈ ∀ ∈
(cid:13) (cid:13) (cid:8) (cid:9)
(cid:13) (cid:13)