Post on 26-Feb-2018
7/25/2019 Conway Funct Analysis
1/418
7/25/2019 Conway Funct Analysis
2/418
Graduate Texts in Mathematics 96
Editorial Board
F. W. Gehring
P.
R. Halmos (Managing Editor)
C. C. Moore
7/25/2019 Conway Funct Analysis
3/418
Graduate Texts in Mathematics
TAKE Un/ZARING. Introduction to Axiomatic Set Theory. 2nd ed.
2 OXTOBY. Measure and Category. 2nd ed.
3 SCHAEFFER. Topological Vector Spaces.
4
HILTON/STAMM BACH.
A Course in Homological Algebra.
5 MACLANE. Categories for the Working Mathematician.
6 HUGHES/PIPER. Projective Planes.
7
SERRE. A Course in Arithmetic.
8 TAKEcn/ZARING. Axiometic Set Theory.
9 HUMPHREYS. Introduction to Lie Al) ebras and Representation Theory.
10 COHEN. A Course in Simple Homotopy Theory.
11
CONWAY.
Functions
of
One Complex Variable. 2nd ed.
12
BEALS.
Advanced Mathematical Analysis.
13
ANDERSON/FuLI.ER. Rings and Categories of Modules.
14 GOLUBITSKy/GuILLFMIN. Stable Mappings and Their Singularities.
15 BERBERIAN.
Lectures
in
Functional Analysis and Operator Theory.
16
WINTER.
The Structure of Fields.
17 ROSENBLATT.
Random Processes. 2nd ed.
18 HALMos. Measure Theory.
19
HALMos.
A Hilbert Space Problem Book. 2nd ed., revised.
20 HUSEMOLLER. Fibre Bundles. 2nd ed.
21
HUMPHREYS. Linear Algebraic Groups.
22
BARNES/MACK.
An Algebraic Introduction to Mathematical Logic.
23
GREUB.
Linear Algebra. 4th ed.
24 HOLMES. Geometric Functional Analysis and its Applications.
25 HEWITT/STROMBERG. Real and Abstract Analysis.
26
MANES.
Algebraic Theories.
27
KELLEY.
General Topology.
28
ZARISKI/SAMUEL.
Commutative Algebra. Vol.
l.
29
ZARISKUSAMUEL.
Commutative Algebra. Vol. 11.
30
JACOBSON.
Lectures
in
Abstract Algebra
I:
Basic Concepts.
31 JACOBSON.
Lectures in Abstract Algebra
11:
Linear Algebra.
32
JACOBSON.
Lectures in Abstract Algebra Ill: Theory of Fields and Galois Theory.
33 HIRSCH.
Differential Topology.
34 SPITZER. Principles of Random Walk. 2nd ed.
35
WERMER.
Banach Algebras and Several Complex Variables. 2nd ed.
36 KELLEy/NAMIOKA et al. Linear Topological Spaces.
37
MONK.
Mathematical Logic.
38
GRAUERT/FRITZSCHE.
Several Complex Variables.
39
ARYESON.
An Invitation to C*-Algebras.
40 KEMENy/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed.
41 APOSTOL.
Modular Functions and Dirichlet Series in Number Theory.
42 SERRE. Linear Representations of Finite Groups.
43
GILLMAN/JERISON.
Rings
of
Continuous Functions.
44 KENDIG. Elementary Algebraic Geometry.
45
LOEYE.
Probability Theory I. 4th ed.
46 LOEYE. Probability Theory 11. 4th ed.
47
MOISE.
Geometric Topology in Dimensions 2 and 3.
continued
after Index
7/25/2019 Conway Funct Analysis
4/418
John
B.
Conway
A Course
in Functional Analysis
Springer Science+Business Media, LLC
7/25/2019 Conway Funct Analysis
5/418
John B. Conway
Department
of Mathematics
Indiana University
Bloomington, IN 47405
U.S.A.
Editorial Board
P.
R. Halmos
Managing Editor
Department
of
Mathematics
Indiana University
Bloomington, IN 47405
U.S.A
F. W. Gehring
Department of
Mathematics
University
of
Michigan
Ann Arbor,
MI
48109
U.S.A.
AMS Classifications: 46-01, 45B05
Library
of
Congress Cataloging in Publication
Data
Conway, John
B.
A course in functional analysis.
(Graduate texts in mathematics: 96)
Bibliography: p.
Includes index.
1. Functional analysis.
I.
Title.
QA320.C658 1985 515.7
With 1 illustration
II. Series.
84-10568
1985 by Springer Science+Business
Media
New York
Originally published
by Springer-Verlag New York
Inc. in
1985
Softcover reprint of
the
hardcover I
st
edition
1985
c. C. Moore
Department of
Mathematics
University
of
California
at Berkeley
Berkeley, CA 94720
U.S.A.
All rights reserved. No part of this book may be translated or reproduced in any
form without written permission from Springer Science+Business
Media, LLC .
Typeset by Science Typographers, Medford, New York.
9 8 7 6 5 4 3 2 1
ISBN 978-1-4757-3830-8 ISBN 978-1-4757-3828-5 (eBook)
DOI 10.1007/978-1-4757-3828-5
7/25/2019 Conway Funct Analysis
6/418
For Ann (of course)
7/25/2019 Conway Funct Analysis
7/418
Preface
Functional analysis has become a sufficiently large area of mathematics that
it is possible to find two research mathematicians,
both
of whom call
themselves functional analysts, who have great difficulty understanding the
work of the other. The common thread is the existence of a linear space with
a topology or two (or more). Here the paths diverge in the choice of how
that topology is defined and in whether to study
the
geometry of the linear
space, or the linear operators on the space, or both.
In this book I have tried to follow the common thread rather than any
special topic. I have included some topics that a few years ago might have
been thought of as specialized but which impress me as interesting and
basic. Near the end of this work I gave into my natural temptation and
included some operator theory that, though basic for operator theory, might
be
considered specialized by some functional analysts.
The word "course" in the title of this book has two meanings. The first is
obvious. This book was meant as a text for a graduate course in functional
analysis. The second meaning is that the book attempts to take an excursion
through many of the territories that comprise functional analysis. For this
purpose, a choice of several tours is offered the
reader-whether
he is a
tourist or a student looking for a place of residence. The sections marked
with an asterisk are not (strictly speaking) necessary for the rest of the book,
but will offer the reader an opportunity to get more deeply involved in the
subject at hand, or to see some applications to other parts of mathematics,
or, perhaps,
just
to see some local color. Unlike many tours, it is possible to
retrace your steps and cover a starred section after the chapter has been left.
There are some parts of functional analysis that are not on the tour. Most
authors have to make choices due to time and space limitations, to say
nothing of the financial resources of our graduate students. Two areas that
7/25/2019 Conway Funct Analysis
8/418
Vlll
Preface
are only briefly touched here,
but
which constitute entire areas by them
selves, are topological vector spaces and ordered linear spaces. Both are
beautiful theories and both have books which do them justice.
The
prerequisites for this book are a thoroughly good course in measure
and integration-together with some knowledge of point set topology. The
appendices contain some of this material, including a discussion of nets in
Appendix
A. In
addition, the reader should
at
least be taking a course in
analytic function theory at the same time that he is reading this book. From
the beginning, analytic functions are used to furnish some examples, but it
is only in the last half of this text that analytic functions are used in the
proofs of the results.
It has been traditional that a mathematics book begin with the most
general set
of
axioms and develop the theory, with additional axioms added
as the exposition progresses. To a large extent I have abandoned tradition.
Thus
the first two chapters are
on
Hilbert space, the third is on Banach
spaces, and the fourth is on locally convex spaces.
To
be sure, this causes
some repetition (though not as much as I first thought it would) and the
phrase" the proof is
just
like the proof
of . . .
" appears several times. But I
firmly believe that this order
of
things develops a better intuition in the
student. Historically, mathematics has gone from the particular to the
general-not
the reverse. There are many reasons for this, but certainly one
reason is
that
the human mind resists abstraction unless it first sees the need
to abstract.
I have tried to include as many examples as possible, even if this means
introducing without explanation some other branches of mathematics (like
analytic functions, Fourier series, or topological groups). There are,
at
the
end
of
every section, several exercises of varying degrees of difficulty with
different purposes in mind. Some exercises just remind the reader that he is
to supply a proof of a result in the text; others are routine, and seek to fix
some
of
the ideas in the reader's mind; yet others develop more examples;
and
some extend the theory. Examples emphasize
my
idea about the nature
of
mathematics and exercises stress my belief that doing mathematics
is
the
way to learn mathematics.
Chapter
I discusses the geometry of Hilbert spaces and Chapter II begins
the theory of operators on a Hilbert space. In Sections 5-8 of Chapter II,
the complete spectral theory of normal compact operators, together with a
discussion
of
multiplicity, is worked out. This material is presented again in
Chapter
IX, when the Spectral Theorem for bounded normal operators is
proved. The reason for this repetition is twofold. First, I wanted to design
the book to be usable as a text for a one-semester course. Second, if the
reader understands the Spectral Theorem for compact operators, there will
be
less difficulty in understanding the general case and, perhaps, this will
lead to a greater appreciation of the complete theorem.
Chapter I I I
is
on
Banach spaces. I t has become standard to do some of
this material in courses on Real Variables. In particular, the three basic
7/25/2019 Conway Funct Analysis
9/418
Preface
ix
principles,
the
Hahn-Banach Theorem, the Open Mapping Theorem, and
the
Principle
of
Uniform Boundedness, are proved.
For
this reason I
contemplated
not
proving these results here, but in the end decided that
they should
be
proved. I did bring myself to relegate to the appendices the
proofs
of
the representation of the dual of
LP
(Appendix
B)
and the dual
of
C
o
(
X)
(Appendix C).
Chapter
IV hits the bare essentials
of
the theory of locally convex spaces
-enough to rationally discuss weak topologies.
I t
is shown in Section 5 that
the
distributions are the dual
of
a locally convex space.
Chapter
V treats the weak and weak-star topologies. This is one of my
favorite topics because of the numerous uses these ideas have.
Chapter
VI looks at bounded linear operators on a Banach space.
Chapter
VII introduces the reader to Banach algebras
and
spectral theory
and
applies this to the study
of
operators
on
a Banach space. It is
in
Chapter
VII that the reader needs to know the elements of analytic function
theory, including Liouville's Theorem and Runge's Theorem. (The latter is
proved using the
Hahn-Banach
Theorem in Section IlLS.)
When
in
Chapter VIII the notion
of
a C*-algebra is explored, the
emphasis
of
the book becomes the theory
of
operators
on
a Hilbert space.
Chapter
IX
presents the Spectral Theorem and its ramifications. This is
done
in
the framework of a C*-algebra. Classically, the Spectral Theorem
has been thought
of
as a theorem about a single normal operator. This it is,
but it
is more. This theorem really tells us about the functional calculus for
a normal operator and, hence, about the weakly closed C*-algebra gener
ated by
the normal operator.
In
Section IX.S this approach culminates in
the
complete description
of
the functional calculus for a normal operator. In
Section IX.lO the multiplicity theory
(a
complete set
of
unitary invariants)
for normal operators is worked out. This topic is too often ignored in books
on
operator theory. The ultimate goal
of
any branch
of
mathematics is to
classify and characterize, and multiplicity theory achieves this goal for
normal operators.
In Chapter
X unbounded operators
on
Hilbert space are examined. The
distinction between symmetric
and
self-adjoint operators is carefully delin
eated
and
the Spectral Theorem for unbounded normal operators is ob
tained as a consequence
of
the bounded case. Stone's Theorem
on
one
parameter unitary groups is proved and the role
of
the Fourier transform in
relating differentiation and multiplication is exhibited.
Chapter
XI, which does not depend on Chapter X, proves the basic
properties of the Fredholm index. Though it is possible to do this in the
context
of
unbounded operators between two Banach spaces, this material is
presented for bounded operators
on
a Hilbert space.
There are a few notational oddities. The empty set is denoted by D. A
reference
number
such as (S.lO) means item number 10
in
Section S
of
the
present chapter. The reference (IX.S.lO) is to (S.lO)
in
Chapter IX. The
reference (A.1.l) is to the first item in the first section
of
Appendix A.
7/25/2019 Conway Funct Analysis
10/418
x
Preface
There are many people who deserve my gratitude in connection with
writing this book. In three separate years I gave a course based on an
evolving set of notes that eventually became transfigured into this book. The
students in those courses were a big help. My colleague Grahame Bennett
gave me several pointers in Banach spaces. My ex-student Marc Raphael
read final versions of the manuscript, pointing out mistakes and making
suggestions for improvement. Two current students, Alp Eden and Paul
McGuire, read the galley proofs and were extremely helpful. Elena Fraboschi
typed the final manuscript.
John B. Conway
7/25/2019 Conway Funct Analysis
11/418
Contents
Preface
CHAPTER
I
Hilbert Spaces
l. Elementary Properties and Examples
2. Orthogonality
3. The Riesz Representation Theorem
4. Orthonormal Sets
of
Vectors and Bases
5. Isomorphic Hilbert Spaces and the Fourier Transform
for the Circle
6. The Direct Sum of Hilbert Spaces
CHAPTER II
Operators on Hilbert Space
l. Elementary Properties and Examples
2. The Adjoint
of
an Operator
3. Projections and Idempotents; Invariant and Reducing
Subspaces
4. Compact Operators
5.* The Diagonalization of Compact Self-Adjoint Operators
6.* An Application: Sturm-Liouville Systems
7.* The Spectral Theorem and Functional Calculus for
Compact Normal Operators
8.* Unitary Equivalence for Compact Normal Operators
CHAPTER III
Banach Spaces
l. Elementary Properties and Examples
2. Linear Operators on Normed Spaces
vii
1
7
11
14
19
24
26
31
37
41
47
50
55
61
65
70
7/25/2019 Conway Funct Analysis
12/418
xii
3. Finite-Dimensional Normed Spaces
4. Quotients and Products of Normed Spaces
5. Linear Functionals
6. The
Hahn-
Banach Theorem
7.* An Application: Banach Limits
8.* An
Application: Runge's Theorem
9.* An Application: Ordered Vector Spaces
1O. The
Dual
of a Quotient Space and a Subspace
11. Reflexive Spaces
12. The Open Mapping and Closed Graph Theorems
13. Complemented Subspaces of a Banach Space
14. The Principle of Uniform Boundedness
CHAPTER
IV
Locally Convex Spaces
l. Elementary Properties and Examples
2. Metrizable and Normable Locally Convex Spaces
3. Some Geometric Consequences of the
Hahn-Banach
Theorem
4.* Some Examples of the Dual Space of a Locally
Convex Space
5.
*
Inductive Limits and the Space of Distributions
CHAPTER
V
Weak Topologies
l. Duality
2. The
Dual
of a Subspace and a Quotient Space
3. Alaoglu's Theorem
4. Reflexivity Revisited
5. Separability and Metrizability
6.
*
An Application: The Stone-Cech Compactification
7. The
Krein-Milman
Theorem
8.
An
Application: The Stone-Weierstrass Theorem
9.* The Schauder Fixed-Point Theorem
10.* The Ryll-Nardzewski Fixed-Point Theorem
11.* An Application: Haar Measure on a Compact Group
12.* The Krein-Smulian Theorem
13.* Weak Compactness
CHAPTER VI
Linear Operators on a Banach Space
l.
The Adjoint of a Linear Operator
2.* The Banach-Stone Theorem
3. Compact Operators
4. Invariant Subspaces
5. Weakly Compact Operators
Contents
71
73
76
80
84
86
88
91
92
93
97
98
102
108
111
117
119
127
131
134
135
138
140
145
149
153
155
158
163
167
l70
175
l77
182
187
7/25/2019 Conway Funct Analysis
13/418
Contents
CHAPTER VII
Banach Algebras and Spectral Theory for
Operators on a Banach Space
1. Elementary Properties and Examples
2. Ideals
and
Quotients
3. The Spectrum
4. The Riesz Functional Calculus
5. Dependence
of
the Spectrum on the Algebra
6. The Spectrum of a Linear Operator
7. The Spectral Theory of a Compact Operator
8. Abelian Banach Algebras
9. * The Group Algebra of a Locally Compact Abelian Group
CHAPTER VIII
C*-Algebras
1. Elementary Properties and Examples
2. Abelian C*-Algebras
and
the Functional Calculus in
C*-Algebras
3. The Positive Elements in a C*-Algebra
4. * Ideals and Quotients for C*-Algebras
5. * Representations
of
C*-Algebras
and
the
Gelfand-Naimark-Segal
Construction
CHAPTER IX
Normal Operators on Hilbert Space
1. Spectral Measures and Representations of Abelian
C*-Algebras
2. The Spectral Theorem
3. Star-Cyclic Normal Operators
4. Some Applications of the Spectral Theorem
5. Topologies
on
J4(.YC')
6. Commuting Operators
7. Abelian von Neumann Algebras
8. The Functional Calculus for Normal Operators:
The Conclusion of the Saga
9. Invariant Subspaces for Normal Operators
1O.
Multiplicity Theory for Normal Operators:
A Complete Set of Unitary Invariants
CHAPTER X
Unbounded Operators
1.
Basic Properties and Examples
2. Symmetric and Self-Adjoint Operators
3. The Cayley Transform
4. Unbounded Normal Operators
and
the Spectral Theorem
XllJ
191
195
199
203
210
213
219
222
228
237
242
245
250
254
261
268
275
278
281
283
288
292
297
299
310
316
323
326
7/25/2019 Conway Funct Analysis
14/418
XIV
5. Stone's Theorem
6. The Fourier Transform and Differentiation
7. Moments
CHAPTER XI
Fredholm Theory
l. The Spectrum Revisited
2. The Essential Spectrum and Semi-Fredholm Operators
3. The Fredholm Index
4. The Components of Yff
5. A Finer Analysis of the Spectrum
APPENDIX A
Preliminaries
l. Linear Algebra
2. Topology
APPENDIX
B
The Dual of LP(fL)
APPENDIX
C
The Dual of Co(X)
Bibliography
List of Symbols
Index
Contents
334
341
349
353
355
361
370
372
375
377
381
384
390
395
399
7/25/2019 Conway Funct Analysis
15/418
CHAPTER
I
Hilbert
Spaces
A Hilbert space is the abstraction of the finite-dimensional Euclidean spaces
of
geometry. Its properties are very regular and contain
few
surprises,
though the presence of an infinity of dimensions guarantees a certain
amount of surprise. Historically, it was the properties of Hilbert spaces that
guided mathematicians when they began to generalize. Some of the proper
ties and results seen in this chapter and the next will be encountered in more
general settings later in this book, or we shall see results that come close to
these but fail to achieve the full power possible in the setting of Hilbert
space.
1.
Elementary Properties and Examples
Throughout this book IF will denote either the real field,
~ ,
or the complex
field, c.
1.1. Definition. I f ' is a vector space over IF, a
semi-inner product
on ' is
a function u: ' X ' IF such that for all
a,
j3 in IF and x, y,
z
in ', the
following are satisfied:
(a)
u(ax +
j3y,
z) = au(x, z) +
j3u(y,
z),
(b)
u(x,
ay
+
j3z)
= iiu(x, y) + jJu(x, z),
(c)
u(
x,
x)
~
0-,,-;------;-
(d)
u(x, y)
=
u(y,
x).
Here, for
a
in IF, i i
= a
if
IF = ~
and i i
is
the complex conjugate of
a
if
IF = c. I f
a
E C, the statement that
a
~ 0 means that
a
E ~ and
a
is
non-negative.
7/25/2019 Conway Funct Analysis
16/418
2
1.
Hilbert Spaces
Note that if
a =
0,
then property (a) implies that
u(O, y)
=
u(a
. 0,
y)
=
au(O,
y)
=
or all
y in 'f.
This
and
similar reasoning shows that for a
semi-inner product u,
(e) u(x,O) = u(O, y) =
for all x, y
in 'f.
In particular,
u(O,
0) =
0.
An
inner product on
'f is a semi-inner product that also satisfies the
following:
(f) I f
u(x,
x) = 0, then x =
0.
An inner product in this book will
be
denoted by
(x, y)
=
u(x,
y).
There is
no
universally accepted notation for an inner product and the
reader will often see
(x, y)
and
(xly)
used in the literature.
1.2. Example. Let
'f be
the collection of all sequences {an: n
~
I}
of
scalars an from IF such that an
=
or all
but
a finite number
of
values
of
n.
I f addition
and
scalar multiplication are defined
on 'f
by
{an} + { P
n
} == {an + P
n
},
a
{
an} == {aa
n
},
then 'f
is a vector space over IF.
I f u({an},{Pn})==L'::=la2n"P2n'
then
u
is a semi-inner product that is
not an inner product. On the other hand,
00
( {an}, {P
n
}) = L an lin ,
n=1
00 1 _
({an}, {P
n
}) = L ;anPn,
n=1
00
({an}, {P
n
})
=
L
n
5
a
n
li
n
,
n=1
all define inner products
on 'f.
1.3. Example. Let (X, J,
p.)
be a measure space consisting of a set
X,
a
a-algebra J of subsets of X, and a countably additive
IR
U {oo} valued
measure p. defined
on J.
I f
f
and
g E L
2(p.)
==
L
2(
X, J,
p.),
then Holder's
inequality implies
f t
E L
1
(p.).
I f
( j ,
g)
=
jftdp.,
then this defines
an
inner product
on
L 2(p.).
Note that Holder's inequality also states that Ifftdp.1 s [flN dp.jl/2 .
[flgl
2
dp.j1/2. This is, in fact, a consequence
of
the following result on
semi-inner products.
7/25/2019 Conway Funct Analysis
17/418
1.1. Elementary Properties and Examples 3
1.4. The Cauchy-Bunyakowsky-Schwarz Inequality. If
( .
.) is
a semi
inner product
on :Y, then
I(x, y)1
2
(x, x)(y,
y)
for all x and y
in
:Y.
PROOF. I f
a E IF and x
and
y E
:Y, then
o (x
-
ay, x
- ay)
=
(x,
x)
-
a(y, x)
- a(x, y)
+ laI
2
(y, y).
Suppose
(y, x)
= be
iO
, b ~
0,
and let a = e-iOt, t in IR.
The
above
inequality
becomes
o
(x,
x)
-
e-iOtbe
iO
-
eiOtbe-
iO
+
t2(y, y)
=
(x,
x)
-
2bt
+ t \ y , y)
= c - 2bt + at
2
== q( t),
where c = (x, x) and a =
(y,
y). Thus q( t) is a quadratic polynomial in
the real variable t and q(t) ~ 0 for all t. This implies that the equation
q(t) = 0 has at
most
one real solution
t.
From the quadratic formula we
find that the discriminant is not positive;
that
is, 0 ~ 4b
2
- 4ac. Hence
o b
2
-
ac
=
I(x, y)1
2
-
(x,
x)(y, y),
proving
the
inequality.
The inequality in (1.4) will
be
referred to as the CBS inequality.
1.5. Corollary. If ( . , .) is a semi-inner product on :Y and
Ilxll
== (x,
X
)1/2
for all x
in :Y,
then
(a)
Ilx
+ yll ~
Ilxll
+
Ilyll for x, yin :Y,
(b)
Ilaxll = lalllxli
for a
in
IF and x
in
:Y.
If
( . ,
.)
is
an
inner product, then
(c)
Ilxll =
0
implies x =
o.
PROOF. The
proofs
of (b) and (c) are left as an exercise. To see (a), note that
for x
and
y
in :Y,
Ilx
+
yll2
= (x +
y,
x + y)
=
IIxl12 +
(y,
x) + (x,
y)
+ IIyl12
=
IIxl12 + 2 Re(x,
y)
+ Ily112.
By
the CBS
inequality, Re(x, y)
~ I(x, y)1
~
IIxIIIIYII.
Hence,
Ilx + yl12 ~ IIxl12 + 211xlillyll +
IIyl12
=
(jlxll + Ilyilf.
The inequality now
follows
by
taking square roots.
7/25/2019 Conway Funct Analysis
18/418
4
1. Hilhert Space,
I f
( . , .)
is a semi-inner product on :r and if x, y E:r, then as was
shown in the preceding proof,
Ilx
+
yl12
=
IIxl12
+
2Re(x ,y )
+
Ily112.
This identity is often called the polar identity.
The quantity
Ilxll
= (x, X
/12
for an inner product ( . , .) is called the
norm of x. I f :r =
IFd (IR
d or C d) and ({ an), {,Bn}) == L ~ ~ l a ) 3 n ' then the
corresponding norm is II {an}
II
= [ L ~ ~ d a n I 2 ] 1 / 2 .
The virtue
of
the norm on a vector space :r is that d(x,
y)
= Ilx - yll
defines a metric on :r [by (1.5)] so that :r becomes a metric space. In fact,
d(x,
y)
= Ilx -
yll
= II(x - z) +
(z -
y)11 ::;; Ilx -
zll
+
liz
-
yll
=
d(x, z)
+ d(z,
y). The other properties of a metric follow similarly.
I f
(
=
IF
d
and
the norm
is
defined as above, this distance function is the usual
Euclidean metric.
1.6. Definition.
A
Hilbert space
is a vector space .Y1' over IF together with
an inner product ( . , .) such that relative to the metric d(x,
y)
= Ilx -
yll
induced
by
the norm, .Y1' is a complete metric space.
I f .Y1'= L 2(fL) and ( I , g) = fft dfL, then the associated norm is Illll =
[flfl2dfLP
/
2.
It
is a standard result of measure theory that
L2(fL)
is a
Hilbert space. I t is also easy to see that IF d is a Hilbert space.
REMARK. The inner products defined on L 2(fL) and IF d are the" usual" ones.
Whenever these spaces are discussed these are the inner products referred
to. The same is true of the next space.
1.7. Example. Let
I
be any set and let [2(1) denote the set of all functions
x:
I
~
IF such that
xU)
= 0 for all but a countable number of
i
and
L
j
E Tlx(i)1
2
f such that (a) 1(0) = 0; (b) for 1
~
k
~
n - 1,
l(k)(t)
exists for all t in [0,1] and I ( k ) is continuous on [0,1]; (c) I ( n
- 1 )
is absolutely
continuous and I(n) E L2(0, 1). For I and
g
in
Yi',
define
7/25/2019 Conway Funct Analysis
21/418
I.2. Orthogonality
7
(b) Show that if
(x + A"',y + A"') == u(x,y)
for all
x
+
A'"
and
y
+
A'"
in
the quotient space
.?r/A"',
then ( . ,
.)
is a
well-defined inner product on
.?r/
A"'.
7. Let Yt' be a Hilbert space over IR and show that there is a Hilbert space : over
C and a map U: Y t ' ~ : such that (a) U is linear; (b) (Uhl'
Uh
2
) =
(hi'
h
2
)
for all hi' h2 in Yt';
(c)
for any k in : there are unique hi'
h2
in Yt' such that
k
=
Uh
l
+
iUh
2
.
(: is called the complexification of
Yt'.)
8. I f
G = {z E C: 0 < Izl < I} show that every J in L ~ ( G ) has a removable
singularity at z =
O.
9. Which functions are in L;(C)?
10. Let G be an open subset of C and show that if a E G, then
{IE
L ~ ( G ) :
J(a) =
O} is closed in L ~ ( G ) .
11. I f {h
n
} is a sequence in a Hilbert space Yt' such that Lnllhnll < 00, then show
that L ' : ~ l h n converges in Yt'.
2. Orthogonality
The greatest advantage of a Hilbert space is its underlying concept of
orthogonality.
2.1. Definition. I f .Yt' is a Hilbert space and I, g E.Yt', then I and g are
orthogonal if U, g)
= O. In symbols,
1.1
g. I f
A, B
~ . Y t ' , then
A .1 B
if
1..1 g for every
I
in
A
and g in B.
I f
.Yt'=
IR
2, this is the correct concept. Two non-zero vectors in
IR
2 are
orthogonal precisely when the angle between them is
7T
/2.
2.2. The
Pythagorean Theorem. II 11,/2"'" In
are pairwise orthogonal
vectors in
.Yt',
then
Ilf1
+
12
+ ... + nl1
2
= 11/1112 +
Ilf2112
+ ... + Il/nl1
2
.
PROOF. I f 11 .1 12' then
Ilf1
+
12112 = U1 + 12'/1
+ 12)
=
Ilfl112 +
2 ReU1'/2)
+
11/2112
by
the
polar
identity. Since
11
.1
12'
this implies the result for
n
=
2.
The
remainder of the proof proceeds by induction and is left to the reader.
Note
that if
I
.1 g, then
I
.1 - g, so Ilf - gl12
=
11.1112
+
Ilg112. The next
result is an easy consequence of the Pythagorean Theorem if
I
and g are
orthogonal,
but
this assumption is not needed for its conclusion.
7/25/2019 Conway Funct Analysis
22/418
8
I. Hilbert Spaces
2.3. Parallelogram Law. If.Yt' is
a Hilbert space
and
f
and g
EO.Yt',
then
Ilf
+ gl12 +
Ilf -
gl12
=
2(llfl1
2
+ IIgI12).
PROOF.
For
any
f
and
g in
.Yt'
the
polar
identity
implies
Now
add. P:
Ilf
+ gl12
=
Ilfll2
+
2 Re(j, g)
+
Ilg11
2
,
Ilf -
gl12
=
Ilfll2
- 2 Re(j, g)
+
Ilgll
2
.
The
next
property of a Hilbert space is truly pivotal. But first we need a
geometric
concept
valid for any vector space over IF.
2.4.
Definition. I f r
is
any
vector space over
IF
and
A
s:;;
r, then
A
is a
convex set
if
for any
x
and
y
in
A
and
$
t
:$
1,
tx
+
(1 - t)y EO
A.
Note
that {tx + (1 - t)
y:
$
t :$
I} is the straight-line segment joining
x and
y. So
a convex set is a set A such that
if
x
and
y
EO A, the entire line
segment joining x and
y is
contained
in
A.
I f r is a vector space, then any linear subspace in r is a convex set. A
singleton set is convex.
The
intersection of any collection of convex sets is
convex. I f .Yt' is a Hilbert space, then every
open
ball B(f;
r)
= {g EO .Yt':
Ilf - gil
h
o
.
By assumption, L(h
o
)
= lim[L(h
n
- h + h
o
)] =
lim[L(h
n
) - L(h) + L(h
o
)] = lim L(h
n
)
- L(h) + L(h
o
).
Hence
L(h) =
limL(h
n
) .
7/25/2019 Conway Funct Analysis
26/418
12
1. Hilbert Spaces
(b)
=
(d): The definition of continuity at 0 implies that L - I ({a ElF:
lal
0 such that
B(O; 8) ~ L -I({a ElF: lal 0: IL(h)1 s cllhll,h
in
X} .
Also, IL(h)1 s IILllllhll for
every
h in X .
PROOF.
Let a
=
inf{ c >
0:
IIL(h)1I s cllhll, h in
X}.
It will be shown that
IILII
=
a;
the remaining equalities are left as an exercise.
I f
f
>
0,
then the
definition of IILII shows that ILlIhll + f)-lh)1 s
IILII.
Hence IL(h)1 s
IILII(lIhll + f). Letting f
~
0 shows that IL(h)1 s IILllllhll for all h. So the
definition of a shows that a s IILII. On the other hand, if IL( h)1 s cllh II
for all
h,
then
IILII
s c. Hence
IILII
s a.
Fix an ho in X and define L:
X ~ IF
by
L(h) = (h, h
o
>.
It is
easy to
see that L is linear. Also, the CBS inequality gives that IL(h)1
= I(h,
ho>1
s IIhllllholi. So L is bounded and IILII s
IIholl.
In fact, L(ho/liholl)
=
(ho/liholl,
ho>
=
IIholl,
so that
IILII
=
IIholi.
The main result of this section
provides a converse to these observations.
3.4. The Riesz Representation Theorem. I f
L:
X ~ IF is a
bounded
linear
functional, then
there
is a
unique
vector ho
in
X such that L(h) = (h, h
o
>
for
every h in X. Moreover,
IILII
=
IIhali.
7/25/2019 Conway Funct Analysis
27/418
I.3. The Riesz Representation Theorem
13
PROOF. Let ,A = ker L. Because
L
is continuous, ,A is a closed linear
subspace of
.:It'.
Since we may assume that ,A
* :It',
,A
1. *
(0). Hence there
is a vector fo in ,A
1.
such that
L(fo)
= 1. Now if h E.:It' and a = L( h),
then
L(h
-
afo)
=
L(h)
-
a
=
0;
so
h
-
L(h)fo
E , A .
Thus
0= (h -
L{h)fo,Jo>
=
(h,Jo> -
L{h)llfoIl
2
.
So if ho =
Ilfoll-%,
L(h) = (h, h
o
> for all h in
.:It'.
I f ~ E.:It' such that (h, h
o
> = (h, h
o
> for all h, then ho - h ~
..l.:lt'.
In
particular,
ho -
h ~
..1
ho
-
h ~ and so ho
=
h
o
. The fact that
IILII =
IIh
o
l
was shown in the discussion preceding the theorem.
3.5. Corollary.
I f
(X,
D,
p.)
is a measure space
and F:
L2(p.)
-> IF
is a
bounded
linear functional, then there is a unique h
0
in L
\ p.)
such that
for every h in L 2(p.).
Of course the preceding corollary
is
a special case of the theorem on
representing bounded linear functionals on
LP(p.),
1
:::;
p
for
every
h in .Jf'. (c) What is the norm
of
the linear functional L defined in (b)?
4.
With the notation as in Exercise 3, define L: .Jf'-'> C by L({ a,,}) =
L ~ ~ l n a " A n - l ,
where IAI < l.
Find
a vector ho in .Jf' such that
L(h)
=
(h,h
o
>
for every h
in .Jf'.
5.
Let .Jf' be the Hilbert space described in Example l.8. I f 0 < t
s
1, define L:
.Jf'-'>
IF
by L(h) = h(t). Show that L is a bounded linear functional, find liLli,
and
find the vector ho in .Jf' such that L( h) = (h, ho
>
or all h in .Jf'.
6. Let .Jf'= L2(0, 1) and let e(l) be the set of all continuous functions on [0,1] that
have a continuous derivative. Let t E [0,1] and define L: e(l)
-'>
IF by L(h) =
h'(t). Show that there
is
no bounded linear functional
on
.Jf' that agrees with
L
on eel).
7/25/2019 Conway Funct Analysis
28/418
14
1. Hilbert Spaces
4. Orthonormal Sets of Vectors and Bases
I t will be shown in this section that, as in Euclidean space, each Hilbert
space can be coordinatized. The vehicle for introducing the coordinates is
an orthonormal basis. The corresponding vectors in IF d are the vectors
{el,eZ, . . . ,e
d
} , where
e
k
is the d-tuple having a 1 in the kth place and
zeros elsewhere.
4.1. Definition. An
orthonormal
subset of a Hilbert space X is a subset
Iff
having the properties: (a) for
e
in Iff, Ilell = 1; (b) if
e
l
,
e
2
E Iff and
e
1
=1= e
2
,
then e
l
..1
e
2
A
basis
for X is a maximal orthonormal set.
Every vector space has a Hamel basis (a maximal linearly independent
set).
The
term
"basis"
for a Hilbert space
is
defined as above and it relates
to the inner product on
X . For
an infinite-dimensional Hilbert space, a
basis is never a Hamel basis. This is not obvious, but the reader will be able
to see this after understanding several facts about bases.
4.2. Proposition. I f
Iff is an orthonormal set in X , then there is a basis for
X
that contains
Iff.
The
proof
of
this proposition is a straightforward application of Zorn's
Lemma and is left to the reader.
4.3. Example. Let
X =
L ~ [0,2'17] and for
n
in
71..
define
en
in X by
en(t) =
(2'17)-I/Zexp(int). Then
{en: n E
7I..} is an orthonormal set in
X .
(Here L ~
[0, 2'17]
is the space of complex-valued square integrable functions.)
It is also true that the set in (4.3) is a basis, but this is best proved after a
bit
of
theory.
4.4. Example.
I f
X = IFd and for 1 ~ k ~ d, e
k
= the d-tuple with 1 in the
kth place and zeros elsewhere, then
{e
l
,
...
,
e
d
}
is a basis for
X .
4.5. Example. Let
X =
12(1) as in Example 1.7.
For
each i in I define e
i
in X by ei(i) = 1 and ei(J) = or j
=1=
i. Then {e
i
: i
E I } is a basis.
The proof of the next result is left as an exercise (see Exercise
5).
It is very
useful
but
the proof is not difficult.
4.6. The Gram-Schmidt Orthogonalization Process. If X
is a Hilbert
space and
{h
n
:
n
E N}
is a linearly independent subset
of X ,
then there
is
an orthonormal set {en: n
EN}
such that for every n, the linear span
of
{e
l
, . ,
en} equals the linear span
of
{hi" '" h
n
}
7/25/2019 Conway Funct Analysis
29/418
IA.
Orthonormal
Sets of
Vectors
and Bases 15
Remember that VA is the closed linear span of
A
(Exercise 2.4).
4.7. Proposition.
Let {e1, . . . ,e
n
} be an orthonormal set in
and
let
A
=
V{
e
l' ...
en}.
I f
P is the orthogonal projection
of
onto
A ,
then
for all
h in
.
n
Ph = ~
(h,ek)e
k
k ~ l
PROOF.
Let Qh = r,Z=l(h, ek)e
k
. I f
1:s;
j :s; n, then (Qh, e
j
) =
r ,Z=/h,ek)(ek ,e)
=
(h ,e ) since ek.L
e
j
for k =F j. Thus (h - Qh,e)
=
0 for 1 :s;
j
:s;
n.
That is,
h
-
Qh
.L A for every
h
in . Since
Qh
is
clearly a vector in A , Qh is the unique vector h 0 in A such that
h
- ho . LA
(2.6). Hence Qh
=
Ph for every h in .
4.8. Bessel's Inequality.
I f
{en:
n
EN}
is an orthonormal set
and
h E
,
then
00
~ I(h, e
n
)12 :s;
IIhll
2
.
n=l
PROOF. Let
h
n
=
h
- r,Z=l(h,ek)e
k
.
Then hn.L
e
k
for 1 :s; k:s;
n
(Why?).
By the Pythagorean Theorem,
IIhll
2
=
IIh n
ll
2
+ ~ l
(h, en)ekr
n
n
;::::
~ I(h, e
k
)1
2
k=l
Since n was arbitrary, the result is proved.
4.9. Corollary.
I f
C
is an orthonormal set in
and
hE
,
then
(h,
e)
=F 0
for
at
most
a countable number
of
vectors e in
C.
PROOF.
For each n;:::: 1 let C
n
= {e E
C: I(h,
e)1 ;::::
l in} . By
Bessel's
Inequality,
C
n
is finite. But U::'=lC
n
= {e E C: (h,e
n
)
=F O}.
4.10. Corollary. I f C is an orthonormal set and h E , then
~ l(h,e)1
2
:s; IIhll
2
.
ef", f
This last corollary is just Bessel's Inequality together with the fact (4.9)
that
at
most a countable number of the terms in the sum differ from zero.
Actually, the sum that appears in (4.10) can be given a better interpreta
t ion-a
mathematically precise one that will be useful later. The question is,
7/25/2019 Conway Funct Analysis
30/418
16
I.
Hilbert Spaces
what
is meant
by
I:{ hi:
i
E I } if hi E.Yf' and I is an infinite, possibly
uncountable, set? Let
:7 be the collection of all finite subsets of
I and
order
:7
by inclusion, so :7 becomes a directed set.
For
each F in :7, define
hF= L { h
i
:
i E F}.
Since this is a finite sum, hF is a well-defined element of
.Yf'.
Now {h F:
FE:7} is a net in .Yf'.
4.11. Definition. With the notation above, the sum I:{ hi: i
E
I} converges
if the net {h
F: F E
:7} converges; the value of the sum is the limit of the
net.
I f
.Yf'=
IF,
the definition above gives meaning to an uncountable sum of
scalars. Now Corollary 4.10 can be given its precise meaning; namely,
I:{I(h,e)1
2
: e E
6"} converges and the value::; IIhl1
2
(Exercise 9).
I f the set
I
in Definition
4.11
is countable, then this definition of
convergent
sum
is
not
the usual one. That is, if {h
n}
is a sequence in .Yf',
then the convergence of I:{ h n:
n EN}
is not equivalent to the convergence
of
: r : ~ l h
n
The former concept of convergence is that defined in (4.11) while
the latter means that the sequence { I : k ~ l h d r : ~ l converges. Even if .Yf'= IF,
these concepts do not coincide (see Exercise 12).
If,
however, I:{ h
n:
n
EN}
converges,
then
I : r : ~ l h n converges (Exercise 10). Also see Exercise 11.
4.12. Lemma. I f C is an orthonormal set and h E.Yf', then
L{(h ,e )e :
e E C }
converges in
.Yf'.
PROOF. By (4.9), there are vectors
e
1
,
e
2
,
. . . in C such that {e E
C:
(h,e) *" O} = {e
1
,e
2
, . . .
}.
W e a l s o k n o w t h a t I : r : ~ 1 1 ( h , e n ) 1 2 : : ; IIhl1
2
0, there is
an
N such that I : r : ~ N I ( h , e n ) 1 2 < ,,2. Let
Fa =
{e
1
,
...
, eN-d
and
let :7= all the finite subsets of C. For F in
:7
define
h
F
==
L{
(h, e)e:
e
E F}. I f
F
and
G
E:7
and both
contain
Fa,
then
IIh
F
- hGII2 = L{I(h,e)1
2
:
e
E (F\G)
U(G\F)}
00
< e
2
.
So {h
F: F E :7} is
a Cauchy net in
.Yf'.
Because .Yf' is complete, this net
converges. In fact, it converges to
I : r : ~ l ( h , en)e
n
4.13. Theorem.
I f
C
is an orthonormal set in
.Yf',
then the following
statements
are equivalent.
(a) C is a basis for .Yf'.
(b)
I f
h E.Yf' and h
.1
C, then h =
0.
(c) V C= .Yf'.
7/25/2019 Conway Funct Analysis
31/418
1.4. Orthonormal Sets of Vectors and Bases
(d) I f
h
E.Yl',
then h
=
L{
(h, e)e:
e
E O"}.
(e) I f g and h E.Yl', then
(g,h)
=
L{(g ,e ) (e ,h) :
eEO"}.
17
(f) I f h E.Yl', then IIhl1
2
= L{I(h, e)12: e E O"} (Parseval's Identity).
PROOF.
(a) =* (b): Suppose h .1. 0" and h =f- 0; then
O"U{
h/llhll} IS an
orthonormal set that properly contains 0", contradicting maximality.
(b)
=
(c): By Corollary 2.11,
VO"=.Yl'
if and only if
0".1
= (0).
(b)
=*
(d):
I f h
E.Yl', then
f= h
- L{(h,e)e:
e
E
O"}
is a well-defined
vector by Lemma 4.12. I f e
1
E
0",
then
(I ,
e
1
) =
(h,
e
1
) - E{ (h,
e)(
e, e
1
) :
e EO"} = (h, e
1
)
-
(h, e
1
) = O. That is, f
EO".1
. Hence f = O. (Is every
thing legitimate in that string of equalities? We don't want any illegitimate
equalities.)
(d) =* (e): This is left as an exercise for the reader.
(e) =*
(f):
Since
IIhl1
2
=
(h,
h), this is immediate.
(f) =* (a): I f 0" is not a basis, then there is a unit vector eo (Ileoll =
1)
in
.Yl' such that eo.l.. 0". Hence, 0 = E{I(e
o
,e)1
2
: e E O"}, contradicting (f) .
Just as in finite-dimensional spaces, a basis in Hilbert space can be used
to define a concept of dimension.
For
this purpose the next result is pivotal.
4.14. Proposition. If .Yf' is a Hilbert space, any two bases have the same
cardinality.
PROOF. Let 0" and
.f7
be two bases for
.Yf'
and put e = the cardinality of
0",
TJ = the cardinality of .f7. I f e or TJ is finite, then e =
TJ
(Exercise 15).
Suppose both e and
TJ
are infinite.
For
e in
0",
let
~ ={ f
E.f7: (e, f )
=f
O}; so ~ is countable. By (4.13b), each f in .f7 belongs to at least one set
~ , e
in
0".
That
is, .f7=
U { ~ :
e
E
O"}.
Hence
TJ
~
e .
~ o
=
e.
Similarly,
e
~
TJ.
4.15. Definition.
The
dimension
of a Hilbert space is the cardinality of a
basis and is denoted by dim.Yf'.
I f (X,
d) is a metric space that is separable and
{B;
=
B(x
i
; e;):
i
E I} is
a collection
of
pairwise disjoint open balls in X, then
I
must be countable.
Indeed, if D is a countable dense subset of
X,
Bi () D =f- 0 for each i in I.
Thus there is a point Xi in Bi ()
D.
So {Xi: i E I} is a subset of
D
having
the cardinality of
I;
thus
I
must be countable.
4.16. Proposition.
If.Yf'
is an infinite-dimensional Hilbert space, then .Yl' is
separable if
and
only if dim.Yf'= ~ o'
7/25/2019 Conway Funct Analysis
32/418
18 I. Hilbert Spaces
PROOF. Let g be a basis for
.Yf'. I f e
l
,
e
2
E g, then lie
l
- e211
2
=
IIedl
2
+
IIe
2
2
= 2. Hence
{B(e;
1/v1): e
E g}
is a collection of pairwise disjoint
open balls in .Yf'. From the discussion preceding this proposition, the
assumption that
.Yf'
is
separable implies g is countable. The converse is an
exercise.
EXERCISES
1.
Verify the statements in Example
4.3.
2. Verify the statements in Example 4.4.
3. Verify the statements in Example 4.5.
4.
Find an
infinite orthonormal set in the Hilbert space of Example 1.8.
5. Using the notation of the Gram-Schmidt Orthogonalization Process, show that
up
to scalar multiple e
l
=
hl/llhdl
and for n
~ 2, en
=
Ilh
n
-
Inll-l(h
n
- /,,),
where
In
is the vector defined formally by
-1 l
h l ~ h l )
In
= de t
det[
(hi '
) r j ~ l (hI' h
n
-
l
)
hI
In
the next three exercises, the reader is asked to apply the Gram-Schmidt
Orthogonalization Process to a given sequence in a Hilbert space. A reference for
this material is pp.
82-96
of Courant and Hilbert
[1953].
6. I f
the sequence
1, x, X 2, .
. . is orthogonalized in L
2
( - 1, 1),
the sequence
en(x) = [t(2n + 1)]1/2P
n
(x)
is
obtained, where
Px) =
_ l _ ( ~ ) n
(x
2
-
1)"
n 2nn dx .
The functions P
n
(x)
are called Legendre polynomials.
7.
I f
the sequence
e-
x
'
/2, xe-
x
2
/2,
x
2
e-
x2
/2, ...
is orthogonalized in
L2(
-
00,
(0),
the sequence en (x) = [2n
n y';
]-1/2
H"
(x )e-
X2
/2
is
obtained,
where
Hn(x) = (-1)" ex2( ~ ) ne-x2.
The functions Hn are Hermite polynomials and satisfy
H;(x) =
2nH,,_
1(x).
8.
I f the sequence e-
x
/
2
, xe-
x
/
2
, x
2
e-
x
/
2
, . . . is orthogonalized in L
2
(O, (0), the
sequence en(x) =
e-
x
/
2
L
n
(x)/n
is
obtained, where
Ln(x)
=
e
x
( ~ r
(xne
7/25/2019 Conway Funct Analysis
33/418
1.5.
Isomorphic Hilbert Spaces and the Fourier Transform for the Circle
19
11. I f
{h
n
}
is a sequence in a Hilbert space and
L ~ ~ J i l h n l l
< 00, show that
L{ h
n
:
n
EN}
converges in the sense of Definition 4.11.
12. Let {an} be a sequence in
f
and prove that the following statements are
equivalent: (a) L{ an: n
E
N} converges in the sense of Definition 4.11. (b) I f 'IT
is any permutation of N, then L ~ ~ l a , , ( n ) converges (unconditional convergence).
(c)
L ~ ~ l l a n l
Yf'.
Similar such arguments show that the concept
of
"isomorphic" is
an equivalence relation
on
Hilbert spaces.
I t
is also certain that this is the
7/25/2019 Conway Funct Analysis
34/418
20
1.
Hilbert Spaces
correct equivalence relation since an inner product is the essential ingredient
for a Hilbert space and isomorphic Hilbert spaces have the
"same"
inner
product.
One
might object that completeness is another essential ingredient
in
the definition of a Hilbert space.
So
it
is
However, this too is preserved
by
an isomorphism. An
isometry
between metric spaces
is
a map that
preserves distance.
5.2. Proposition. I f V: ~ X
is
a linear map between Hilbert spaces, then
V is an isometry
if
and only if
(Vh,
Vg)
= (h,
g) for all h,
g in .
PROOF. Assume (Vh, Vg) =
(h,
g) for all h, g in . Then II Vhl1
2
=
(Vh, Vh) = (h, h) =
IIhl1
2
and V is an isometry.
Now
assume that
V
is an isometry.
I f
h,
g
E
and
;\.
E
IF,
then
Ilh + ;\.g112 = II Vh +
;\.VgI1
2
. Using the polar identity on both sides of this
equation gives
IIhl1
2
+ 2ReX(h,g) + 1;\.1
2
11g11
2
= IIVhl1
2
+ 2ReX(Vh,Vg) + 1;\.1
2
1IVgI1
2
.
But
II
Vhll =
Ilhll and II Vgll
=
Ilgll, so this equation becomes
ReX(h, g)
=
ReX(Vh, Vg)
for any ;\. in IF.
I f
IF = IR, take ;\. = 1. I f IF = C, first take ;\. = 1 and then
take ;\. = i to find that (h, g) and (Vh, Vg) have the same real and
imaginary parts.
Note
that an isometry between metric spaces maps Cauchy sequences into
Cauchy sequences. Thus an isomorphism also preserves completeness.
That
is, if an inner product space is isomorphic to a Hilbert space, then it must be
complete.
5.3. Example. Define S: 12 ~ 12 by
S(
0'1,0'2' . . . ) = (0,0'1,0'2' ... ). Then
S is an isometry that is not surjective.
The
preceding example shows that isometries need not be isomorphisms.
A word about terminology. Many call what
we
call an isomorphism a
unitary operator.
We shall define a unitary operator as a linear transforma
tion U: Y l ' ~ Yl' that is a surjective isometry. That is, a unitary operator is
an isomorphism whose range coincides with its domain. This may seem to
be a minor distinction, and in many ways it is. But experience has taught me
that there is some benefit in making such a distinction, or at least in being
aware of it.
5.4. Theorem. Two Hilbert spaces are isomorphic
if
and only
if
they have the
same dimension.
PROOF. I f
U:
Y l ' ~
X is
an isomorphism and I
is
a basis for
,
then it
is
easy to see that UI == {Ue:
eEl }
is a basis for X . Hence, dim
=
dim X .
7/25/2019 Conway Funct Analysis
35/418
I.5. Isomorphic Hilbert Spaces and the Fourier Transform for the Circle
21
Let Yt' be a Hilbert space and let
g
be a basis for Yt'. Consider the
Hilbert space
l2(g).
I f h
EYt',
define h:
g ~ IF
by h(e) =
(h,e). By
Parseval's Identity h E 12(
g)
and
Ilhll
=
IIhll.
Define
U: Y t ' ~ 12(
g) by
Uh
=
h.
Thus
U
is linear and an isometry.
It
is easy to see that
ranU
contains all the functions f in
l2(
g) such that f( e) =
or all but a finite
number of e; that is, ranU is dense. But
U,
being an isometry, must have
closed range. Hence
U: Y t ' ~ 12( g)
is an isomorphism.
I f f is a Hilbert space with a basis %, f is isomorphic to 12(%).
I f
dim Yt'= dim
f, g
and
%
have the same cardinality; it is easy to see that
12( g) and 12( % ) must be isomorphic. Therefore Yt' and f are isomorphic
5.5.
Corollary.
All
separable infinite dimensional Hilbert spaces are isomor
phic.
This section concludes with a rather important example of an isomor
phism, the Fourier transform on the circle.
The
proof of the next result can be found as an Exercise on p.
263
of
Conway [1978]. Another proof will be given later in this book after the
Stone-Weierstrass Theorem is proved.
So
the reader can choose to assume
this for the moment. Let U} = {z
E
C:
Izi
< I}.
5.6. Theorem. I f f:
aU} ~
C
is
a continuous function, then there
is
a
sequence {Pn(z,
i )} of
polynomials
in
z and
z such
that Pn(z,
z) ~ fez)
uniformly
on aU}.
Note that if
z
E aU},
z
= z-l. Thus a polynomial in
z
and
z
on
aU}
becomes a function of the form
I f
we put
z
=
e
iO
,
this becomes a function of the form
Such functions are called trigonometric polynomials.
We can now show that the orthonormal set in Example 4.3 is a basis for
L ~ [ O ,
27T]. This is a rather important result.
5.7. Theorem.
If
for each n
in
71..,
en(t)
==
(27T)-1/2
exp(int), then {en:
nElL} is
a basis for
L ~ [ O , 27T].
PROOF. Let .07= O : : : Z ~ - n < X k e k :
7/25/2019 Conway Funct Analysis
36/418
22
I. Hilbert Spaces
is
'??= {IE
Cdo,
277"): f(O) = f(277")}.
To do this, let
fE Cf/
and define
F:
o[)) ~ C by F(e
it
)
= f(t).
F is continuous. (Why?) By (5.6) there is a
sequence
of
polynomials in z and z, {Pn(z, Z)}, such that Pn(z, z) ~ F(z)
uniformly
on
O[)).
Thus Pn(eil,e-
it
)
-->
f(t)
uniformly on
[0,2'1T].
But
Pn(
e
it
, e-
it
) E:Y.
Now
the closure of Cf/ in L ~ [0,277") is all of L ~ [0,277") (Exercise 6). Hence
V{e
n
: n E
Z}
= L ~ [ 0 , 2 7 7 " )
and
{en}
is thus a basis (4.13).
Actually, it is usually preferred to normalize the measure on [0,277"). That
is, replace
dt
by (277")
-1 dt, so
that the total measure of
[0,277") is 1.
Now
define
en(t)
= exp(int). Hence {en:
n E Z}
is a basis for .Yl'=
L ~ ( [ 0 , 2 7 7 " ) , ( 2 7 7 " ) - l d t ) . I f
fE.Yl', then
5.S
is called the nth Fourier coefficient of f, n in Z. By (5.7) and (4.13d),
00
5.9
n= - 00
where this infinite series converges to f in the metric defined by the norm of
.Yl'. This is called the Fourier series
of
f.
This terminology is classical and
has been adopted for a general Hilbert space.
I f .Yl'
is any Hilbert space and
iff is
a basis, the scalars
{< h, e); e
E
iff}
are called the Fourier coefficients of h (relative to iff) and the series in
(4.13d) is called the Fourier expansion
of
h (relative to iff).
Note
that
Parseval's Identity applied to (5.9) gives that L : ; ' ~
_ool/(n)1
2
L2(X, a, ""1) EB
L
2
(
X, a, lL2) defined by VI
=
11
EB
12' where
fj
is the equivalence class of
L
2
(X,
a,,.,,)
corresponding to
I,
is well defined, linear, and injective. Show that
U
is an isomorphism iff
""1
and ""2 are mutually singular.
7/25/2019 Conway Funct Analysis
40/418
CHAPTER
II
Operators on Hilbert Space
A large area
of
current research interest is centered around the theory
of
operators on Hilbert space. Several other chapters in this book will be
devoted to this topic.
There
is a marked contrast here between Hilbert spaces and the Banach
spaces that are studied in the next chapter. Essentially all of the information
about the geometry of Hilbert space is contained in the preceding chapter.
The geometry of Banach space lies in darkness
and
has attracted the
attention
of
many
talented research mathematicians. However, the theory of
linear operators (linear transformations) on a Banach space has very few
general results, whereas Hilbert space operators have an elegant and well
developed general theory. Indeed, the reason for this dichotomy is related to
the opposite status of the geometric considerations. Questions concerning
operators on Hilbert space
don't
necessitate
or
imply any geometric difficul
ties.
In addition to the fundamentals of operators, this chapter will also
present an interesting application to differential equations in Section
6.
1. Elementary Properties and Examples
The
proof of
the next proposition is similar to that of Proposition 1.3.1 and
is left
to
the reader.
1.1. Proposition. Let yt> and f be Hilbert spaces and A: yt> f a linear
transformation. The following statements are equivalent.
(a) A is continuous.
(b) A is continuous at O.
7/25/2019 Conway Funct Analysis
41/418
ILL
Elementary Properties and Examples
27
(c)
A is continuous at some point.
(d) There is a constant
c
>
0
such that
IIAhl1 ~ cllhll
for all h
in .Yf'.
As
in
(1.3.3),
if
IIA
II = sup{IIAhll: h E
.Yf',
Ilhll
~ I},
then
IIAII = sup{IIAhll:
IIhll
= I}
= sup{IIAhll/llhll:
h
*
O}
= inf{ c
> 0: IIAhl1 ~
cllhll, h in
.Yf'}.
Also,
IIAhl1
~
IIAllllhll.
IIAII
is called the
norm
of
A
and
a linear transfor
mation with
finite norm is called
bounded. Let
Jj(.Yf', g) be
the
set of
bounded linear transformations from .Yf' into g. For.Yf'=
g, Jj(.Yf',.Yf')
== Jj(.Yf'). Note that Jj(.Yf',
0=)
= all the bounded linear functionals on .Yf'.
1.2. Proposition.
(a) If A and
B E
Jj(.Yf',
g) ,
then A + B
E
Jj(.Yf',
g) ,
and
IIA + BII ~ IIAII + IIBII
(b)
I f
a
E
0= and A
E Jj(.Yf',
g ) , then aA
E Jj(.Yf',
g) and IlaA11
=
lalllAII
(c)
I f A
E
Jj(.Yf',
g)
and
B E
Jj(g,
l ') ,
then
BA
E
Jj(.Yf', l')
and
IIBAII
~
IIBIIIIAII
PROOF. Only (c) will be proved; the rest of the proof is left
to
the reader.
I f
kEf, then IIBkl1 ~ IIBllllkll. Hence, if h E.Yf', k = Ah E f and so
IIBAhl1
~ IIBllllAhl1 ~ IIBIIIIAllllhll
By
virtue
of the preceding proposition,
dCA,
B) = IIA -
BII
defines a
metric on Jj(.Yf',
g) .
So it makes sense to consider Jj(.Yf',
g)
as a metric
space. This will
not
be examined closely until
later
in
the
book, but
later
in
this
chapter the idea
of
the
convergence
of
a sequence
of
operators
will
be
used.
1.3. Example.Ifdim.Yf'=n< o o a n d d i m g = m < oo,let{el, . . . ,en}be
an
orthonormal
basis for .Yf' and let {f
l
, . . . , f
m
} be an
orthonormal
basis
for f .
It
can
be shown
that
every linear
transformation
from
.Yf'
into g is
bounded
(Exercise 3).
I f
1
~ j
~ n , 1 ~
i
~
m ,
let aij =
(Ae
i
, f
i
).
Then
the
m X n
matrix (a
i
) represents
A
and every such matrix represents an
element
of Jj(.Yf',
g) .
1.4. Example.
Let
[2 ==
[2(1\1)
and
let
e
l
, e
2
, ...
be its usual basis.
I f
A
E Jj(l2),
form
aij =
(Ae
j
, e
i
) .
The
infinite
matrix ( a
i
)
represents
A
as
finite
matrices
represent
operators on
finite dimensional spaces. However,
this representation has limited value unless the matrix has a special form.
One difficulty is
that
it is
unknown how to
find the
norm
of
A
in terms of
7/25/2019 Conway Funct Analysis
42/418
28
II. Operators
on
Hilbert Space
the entries in the matrix. In fact, if 2 < n < 00, there is no known formula
for the norm of a matrix in terms of its entries. A sufficient condition that is
useful is known, however (see Exercise 11).
1.5. Theorem. Let (X,
D, JL)
be a a-finite measure space and put
:if '=
L 2( X, D, JL) == L 2(JL). I f
0, the a-finiteness of the measure space implies that there is a
set .::1 in
D,
0
f
is a linear
transformation such that L:IIAenll < 00. Show that A is bounded.
4. Proposition 1.2 says that d(A,B) = IIA
- BII
is a metric on ?I(Ji ', f). Show
that ?I(Ji',
f)
s complete relative to this metric.
S.
Show that a multiplication operator
M.p
(1.S) satisfies
M;
=
M.p
if and only if
7/25/2019 Conway Funct Analysis
45/418
II.2. The Adjoint of an Operator 31
2. The Adjoint of an Operator
2.1. Definition. If X and :f are Hilbert spaces, a function
u: Xx:f IF
is a
sesquilinear form
if for
h,
g in X , k,
f
in : f , and
a,
f3 in IF,
(a)
u(ah +
f3g, k)
=
au(h,
k) + f3u(g, k);
(b)
u(h,ak + f3f)
=
au(h, k) + 13u(h,f).
The prefix "sesqui" is used because the function is linear in one variable
but
(for IF = C) only conjugate linear in the other. ("Sesqui" means
" one-and-a-half.")
A sesquilinear form is bounded if there is a constant M such that
lu(h,
k)1
.::;;
Mllhllllkil
for all
h
in X and
k
in
: f .
The constant
M
is
called a bound for
u.
Sesquilinear forms are used to study operators. I f A
E
l(
X ,
X"), then
u(h, k) == (Ah,
k)
is a bounded sesquilinear form. Also, if
B E
leX", X),
u( h, k) ==
(h, Bk)
is a bounded sesquilinear form. Are there any more? Are
these two forms related?
2.2. Theorem. I f u:
Xx:f
IF is
a bounded sesquilinear form with bound
M,
then there are unique operators A in l(
X ,
X") and B
in
l( X",
X )
such
that
2.3
u(h,k) = (Ah,k) = (h,Bk)
for all h in
X and
k
in
:f and IIA
II, IIBII .::;;
M.
PROOF.
Only the existence of
A
will be shown.
For
each
h
in
X ,
define
L
h
:
% ~ IF by Lh(k) = u(h,
k).
Then Lh is linear and ILh(k)1 .::;; Mllhllllkil.
By
the Riesz Representation Theorem there is a unique vector f in X" such
that
(k , j )
=
Lh(k)
= u(h,k) and 11Il1.::;; Mllhll. Let Ah =f .
It
is left as
an exercise to show that
A
is
linear (use the uniqueness part of the Riesz
Theorem). Also, (Ah, k) = (k, Ah) = (k, f) =
u(h, k).
I f Al
E
leX,
% ) and u(h, k) =
(AIh,
k), then
(Ah
- AIh, k) = 0 for
all k; thus Ah - AIh = 0 for all h. Thus, A is unique.
2.4. Definition. If
A
E
l(
X ,
X"),
then the unique operator
B
in
leX",
X )
satisfying (2.3) is called the adjoint of A and is denoted by
B
= A*.
The adjoint of an operator will usually be used for operators in l( X),
rather than
l(
X ,
X"). There is one notable exception.
2.5. Proposition. I f U E l( X ,
: f ) ,
then U is an isomorphism if and only
if
U is invertible and U-
I
= U
*.
PROOF. Exercise.
7/25/2019 Conway Funct Analysis
46/418
32
II. Operators on Hilbert Space
From now
on
we will examine and prove results for the adjoint
of
operators in d( ").
Often, as
in
the next proposition, there are analogous
results for the adjoint
of
operators
in
d( ", .Jf"). This simplification is
justified, however, by the cleaner statements
that
result. Also, the interested
reader will have no trouble formulating the more general statement when it
is needed.
2.6. Proposition.
If A, B E d(") and a E
IF,
then:
(a) (aA + B)*
= aA*
+ B*.
(b)
(AB)* = B*A*.
(c) A** == (A*)*
=
A.
(d)
I f
A is invertible in
d(")
and
A
-1
is its inverse, then
A*
is invertible
and
(A*)
-1 = (A -1)*.
The proof of
the preceding proposition is left as
an
exercise,
but
a word
about part
(d) might
be
helpful. The hypothesis that
A
is invertible
in
d( ") means that there is an operator A -1 in d( ") such that
AA -1 =
A -lA = I. I t is a remarkable fact that if A is only assumed to
be
bijective,
then A
is invertible
in
d(
").
This is a consequence
of
the Open Mapping
Theorem, which will be proved later.
2.7. Proposition.
If A
E
d("), IIAII
=
IIA*II
=
IIA*Alll/2.
PROOF. For h
in ",
Ilhll:s;; 1, IIAhl1
2
= (Ah, Ah) = (A*Ah, h) :s;;
IIA*Ahllllhll :s;; IIA*AII :s;; IIA*llllAII Hence
IIAI12:s;;
IIA*AII :s;;
IIA*IIIIAII
Using the two ends of this string of inequalities gives IIAII :s;; IIA*II when
IIAII is cancelled. But A
=
A** and so if
A*
is substituted for
A,
we get
IIA*II
:s;;
IIA**II =
IIAII Hence
IIAII = IIA*II. Thus
the string
of
inequalities
becomes a string
of
equalities
and
the
proof
is complete.
2.8. Example. Let
(X,
g,
Jl)
be a a-finite measure space
and
let
M
be
the
multiplication operator with
~ m b o l
cp (1.5).
Then M*
is M;p, the multipli
cation operator
with symbol
cpo
I f an
operator
on
IF d is represented by a matrix, then its adjoint IS
represented by the conjugate transpose of the matrix.
2.9. Example. I f
K
is the integral operator with kernel
k
as in (1.6), then
K * is the integral operator with kernel k *(x, y) == k (y, x).
2.10. Proposition.
I f
S: /2 ---> /2
is
defined by S(a
1
, a
2
,
) =
(0,a
1
,a
2
, ), then S
is
an isometry and S*(a
1
,a
2
, )
=
(a
2
,a
3
, .. . ) .
PROOF.
It has
already been mentioned that S is an isometry (1.5.3).
For (a,,)
and (/3
n
) in /2, (S*(a
n
),(/3n
=
a
n
,S(/3n =
a
1
,a
2
, ... ),(0,/31,/32'
7/25/2019 Conway Funct Analysis
47/418
II.2. The Adjoint of an Operator
33
...
= a
2
P
l
+ a
3
P2 +
...
= a
2
,a
3
,
),(f3l,/3
2
,
. Since this holds
for every (f3
n
), the result is proved.
The
operator S in (2.10) is called the
unilateral shift
and the operator S
*
is called the backward shift.
The operation of taking the adjoint of an operator is, as the reader may
have seen from the examples above, analogous to taking the conjugate of a
complex number. I t
is
good to keep the analogy in mind, but do
not
become
too religious about it.
2.11. Definition. I f A
E
JB( .Yt'), then: (a) A is hermitian or self-adjoint if
A* = A; (b) A is normal if AA* = A*A.
In the analogy between the adjoint and the complex conjugate, hermitian
operators become the analogues of real numbers and, by (2.5), unitaries are
the analogues of complex numbers of modulus
1.
Normal operators, as we
shall see, are the true analogues of complex numbers. Notice that hermitian
and unitary operators are normal.
In light of (2.8), every multiplication operator M is normal; M is
hermitian if and only if cf> is real-valued;
M
is unitary if and only if
1cf>1
==
1 a.e. [ILl. By (2.9), an integral operator K with kernel k is hermitian
if
and
only if
k(x,
y)
=
key,
x)
a.e.
[IL
X
ILl.
The unilateral shift is not
normal (Exercise
6).
2.12. Proposition.
I f
.Yt' is a C-Hilbert space and A
E
JB(.Yt'), then A is
hermitian
if
and only if
(Ah, h)
E
R
for all h in
.Yt'.
PROOF. I f A = A*,
then
(Ah, h) = (h,
Ah)
=
(Ah, h); hence
(Ah, h) E
R.
For
the converse, assume
(Ah, h) is
real for every h in .Yt'. I f a E C and
h, g E.Yt', then
(A(h +
ag), h
+
ag)
= (Ah, h) +
ii(Ah,
g) + a(Ag, h)
+
laI
2
(Ag,
g)
E
R. So this expression equals its complex conjugate. Using
the fact that
(Ah, h)
and (Ag,
g) E
R yields
a(Ag, h)
+
ii(Ah,
g) = ii(h, Ag)
+
a(g, Ah)
= ii(A*h, g)
+ a(A*g,
h).
By first taking
a
= 1 and then
a
= i, we obtain the two equations
(Ag, h)
+
(Ah, g) = (A*h, g)
+
(A*g, h),
i(Ag, h)
-
i(Ah, g) =
-i(A*h, g)
+ i(A*g, h).
A little arithmetic implies
(Ag,
h) = (A*g, h), so A = A*.
The
preceding proposition
is
false if it is only assumed that .Yt' is an
R-Hilbert space. For example, if A = [ _ on R 2, then (Ah, h) = 0
7/25/2019 Conway Funct Analysis
48/418
34
II. Operators on Hilbert Space
for all h in 1R2. However,
A*
is the transpose of A
and
so
A* =1=
A. Indeed,
for
any operator
A on an IR-Hilbert space,
(Ah, g)
E
IR.
2.13. Proposition.
If
A
=
A*, then
IIAII
= sup{I(Ah, h)l: Ilhll = I}.
PROOF. Put
M
=
sup{I(Ah,
h)l: Ilhll = I}.
I f
Ilhll =
1,
then I(Ah,
h)1 ::::;
IIAII; hence
M::::; IIAII. On
the
other
hand, if
Ilhll
= Ilgll = 1, then
( A
(h
g),
h g) =
(Ah,
h)
(Ah,
g)
(Ag,
h) + (Ag, g)
= (Ah,
h)
(Ah,
g) (g, A*h) +
(Ag, g).
Since A = A*, this implies
(A{h g),h g) =
(Ah,h)
2Re(Ah,g) + (Ag,g).
Subtracting
one
of these two equations from the other gives
4Re(Ah, g) =
(A{h
+ g), h + g) - (A(h - g), h - g).
Now
it is easy
to
verify that
I(Af,f)1 ::::; MIlf112
for
any
f in . Hence
using the parallelogram law we get
4Re(Ah, g)
::::; M(llh +
gl12
+ Ilh _
g112)
= 2M(llh112 + Ilg11
2
)
=4M
since h a n d g are unit vectors.
Now
suppose
(Ah, g)
= eiol(Ah, g)l.
Replacing
h in the inequality above with e-ioh gives
I(Ah,
g)1 ::::; M if
Ilhll
= Ilgll = 1.
Taking the
supremum
over all g gives IIAhl1
::::;
M when
Ilhll
= 1.
Thus IIAII ::::; M.
2.14. Corollary. If A
=
A* and (Ah, h)
=
0 for all h, then A
= o.
The preceding
corollary is
not
true unless A = A *, as
the
example given
after Proposition
2.12 shows. However, if a complex Hilbert space is
present, this hypothesis
can
be deleted.
2.15.
Proposition. If is
a C-Hilbert space and A
E .'14()
such that
(Ah, h)
=
0
for all h
in
, then A
=
o.
The
proof of
(2.15) is left to
the
reader.
I f
is a
C-Hilbert
space
and
A
E
~ ( ) ,
then B =
(A + A*)j2 and
C
= (A -
A*)j2i are self-adjoint
and
A
=
B
+
iC.
The operators Band
C are called, respectively, the real and imaginary parts of A.
7/25/2019 Conway Funct Analysis
49/418
n.2.
The Adjoint of an Operator
35
2.16. Proposition. I f A E 88() , the following statements are equivalent.
(a)
A is normal.
(b) JJAhJJ
=
IIA*hll
for all h.
I f
is a C-Hilbert space, then these statements are also equivalent to:
(c) The real and imaginary parts
of
A commute.
PROOF. I f h
E
, then JJAhJJ2 - JJA*hJJ2 = (Ah, Ah) -
(A*h,
A*h) =
A*A -
AA*)h, h).
Since
A*A
-
AA*
is hermitian, the equivalence of (a)
and (b) follows from Corollary 2.14.
I f
B,
C are the real and imaginary parts of A, then a calculation yields
A*A
=
B2
-
iCB
+ iBC +
C
2
,
AA*
=
B2
+
iCB
-
iBC
+
C
2
.
Hence
A*A =
AA* if and only if
CB
=
BC,
and so (a) and (c) are
equivalent.
2.17. Proposition. I f A
E
88(
) ,
the following statements are equivalent.
(a) A is an isometry.
(b) A*A = I.
(c)
(Ah, Ag)
=
(h, g)
for all h,
gin .
PROOF. The proof that (a) and (c) are equivalent was seen in Proposition
1.5.2. Note that if
h,
g
E
, then
(A*Ah, g)
= (Ah,
Ag).
Hence (b) and
( c) are easily seen to be equivalent.
2.1S. Proposition.
If
A
E
88(), then the following statements are equiv
alent.
(a) A is unitary.
(b)
A is a surjective isometry.
(c)
A is a normal isometry.
PROOF. (a) = (b): Proposition 1.5.2.
(b) = (c):
By
(2.17),
A*A
=
I.
But it is easy to see that the fact that
A
is
a surjective isometry implies that
A -1
is also. Hence by (2.17) 1=
(A -l)*A -1 = (A*)-lA -1 = (AA*)-\ this implies that A*A = AA* = I.
(c) = (a): By (2.17), A*A = I. Since A is also normal, AA* = A*A = I
and
so
A
is surjective.
We conclude with a very important, though easily proved, result.
2.19. Theorem. I f A E 88(), then kerA = (ranA*)-L.
PROOF.
I f
h
E kerA and g E , then
(h, A*g) = (Ah, g) = 0,
so kerA
c:::;;
(ran
A*)
-L .
On
the other hand, if
h ..1
ran
A*
and g
E
, then
(Ah, g)
=
(h,
A*g) = 0; so (ran
A*) -L
c:::;; ker A.
7/25/2019 Conway Funct Analysis
50/418
36
II. Operators on Hilbert Space
Two facts should be noted. Since A* * = A, it also holds that ker A* =
(ran A) 1- Second, it is not true that (ker
A)
1- = ran A* since ran A* may
not
be
closed. All that can be said is that (kerA)1-= cl(ranA*) and
(ker
A*)
1-
= cl(ran
A).
EXERCISES
1.
Prove Proposition 2.5.
2.
Prove Proposition
2.6.
3. Verify the statement in Example 2.8.
4. Verify the statement in Example 2.9.
5.
Find the adjoint of a diagonal operator (Exercise 1.8).
6. Let S be the unilateral shift and compute
SS*
and S*S. Also compute S"S*"
and S*"S".
7.
Compute the adjoint of the Volterra operator V (1.7) and V + V*. What is
ran(V
+
V*)?
8.
Where was the hypothesis that '
is
a Hilbert space over C used in the proof of
Proposition 2.12?
9.
Suppose
A = B
+
iC,
where
B
and C are hermitian and prove that
B = (A
+
A*)/2,
C
= (A - A*)/2i.
10. Prove Proposition 2.15.
11.
I f
A and B are self-adjoint, show that A B IS self-adjoint if and only if
AB = BA.
12. Let L ~ ~ o a " z " be a power series with radius of convergence
R,
0 < R :0:; 00. I f
A
E
8l(
' )
and
IIA
II
0 as
n
---> 00. I f
BA
= AB, show that BT =
TB.
14. I f fez) =
expz
=
L ~ ~ o z " / n
and A
is
hermitian, show that f(iA) is unitary.
15. I f A is a normal operator on
' ,
show that A is injective if and only if A has
dense range. Give an example of an operator
B
such that ker
B = (0) but
ran
B
is
not dense. Give an example of an operator C such that C
is
sUIjective but
kerC * (0).
16. Let Mq, be a multiplication operator (1.5) and show that kerMq,
=
(0) if and
only if /L({x: CP(x)
=
OJ)
=
O. Give necessary and sufficient conditions on cp
that ran Mq, is closed.
7/25/2019 Conway Funct Analysis
51/418
11.3. Projections and Idempotents; Invariant and Reducing Subspaces
3. Projections and Idempotents; Invariant and
Reducing Subspaces
37
3.1. Definition. An idempotent on is a bounded linear operator E on
such that
E2
= E. A projection is an idempotent P such that ker P
=
(ran P) -L
I f .A :s; , then PJ I is a projection (Theorem 1.2.7).
It
is not difficult to
construct an idempotent that
is not
a projection (Exercise 1).
Let E be any idempotent and set .A = ran E and JV= ker E. Since E is
continuous, JV is a closed subspace of
.