ILBERT SPACE METHODS
. . . . . . . . QUANTUM MECHANICS

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FuNDAMENTAL SciENCES

ILBERT SPACE METHODS
QUANTUM MECHANICS
Werner 0. An1rein

PFL Press
A Swiss academic publisher distributed by CRC Press
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This book is published under the editorial direction of
Professor Philippe-Andre Martin (EPFL, Lausanne).

is an in1print owned by Presses polytechniques et universitaires romandes, a
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Preface

This text is based on lectures given at the University of Geneva during the period

1994- 2005. These courses were intended for advanced undergraduate students who
had completed a first course in quantum mechanics and were thus expected to be familiar with the physical aspects and the basic 1nathernatical formalism of quantum
theory. Partly due to lack of time, quantum mechanics is often taught with no or little
exposition of the mathematical questions arising through the introduction of infinitedimensional Hilbert spaces. I hope that the present volume will prove useful, especially to somewhat theoretically minded students, for deepening their knowledge and
understanding of the Hilbert space aspects of quantum mechanics, and prepare them
for reading research papers.
Mostly the lectures were organised as one-year courses (80 hours plus 25 hours of
problem sessions) and covered essentially the contents of Chapters 1- 5 and parts of
Chapters 6 or 7. To offer a few more applications, some 1naterial has been added to the
original lecture notes (Sections 5.8, 6.6, 6.7, 7.2 and 7.5). Of course a strict selection
of topics and applications to be treated had to be made from the outset. The emphasis is
placed on a certain number of basic mathematical techniques, usually without striving
for the most general results. For example there is no discussion of quadratic forms,
and we have avoided the use of techniques from stochastic analysis. However we give
essentially complete proofs for all results involving Hilbert space objects. Some of
these proofs are collected in appendices to the various chapters. Son1e acquaintance
with measure theory is required: the essential facts are explained without detailed
proofs.
Chapter 1 gives the basic properties of Hilbert space and a description of the necessary material from measure theory. In Chapter 2 we present various classes of bounded
linear operators and general notions on unbounded operators, including the invariance
of self-adjointness under a class of perturbations. The problem of self-adjointness is
further investigated in Chapter 3 which contains the theory of extensions of symmetric
operators, with applications to Sturm-Liouville and Schrodinger operators. Chapter 4
deals with the spectral theory of self-adjoint operators, in particular with the spectral
theorem and with the various spectral types. In Chapter 5 we prove Stone's Theorem and then discuss the fundamental aspects of scattering theory: scattering states,
asymptotic condition and wave operators, S-matrix, scattering cross sections. Chapter
6 is devoted to the Mourre method for controlling the resolvent of self-adjoint operators near the real axis and to implications for their spectrum. In the final Chapter 7
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PREFACE

we present stationary-state scattering theory and various applications of the results of
Chapter 6: asymptotic completeness, properties of the S-matrix, time delay and the
Flux-Across-Surfaces Theorem.
Each chapter ends with some bibliographical notes and a selection of problems.
The bibliography consists mostly of books. They are referred to for alternative or more
advanced presentations of some material or for certain points not treated in the present
text. A very small number of original and review papers are cited for those interested
a deeper understanding of certain topics treated in the text. In combination with
tnodern electronic means. these papers can also be useful as a basis for searching in
the vast literature.
The majority of the problems have been tested in class-room sessions. Many of
then1 are meant to help students to become familiar with the concepts discussed in the
main body; son1e require a more detailed study of technical aspects of the text. A few
of the more difficult problems are provided with a hint for the solution.
As regards notations, it should be pointed out that some symbols have more than
one 1neaning in the text. In particular: the symbol II · II is used for various norms, the
Greek letter CJ for spectra and for scattering cross sections, the letter P for projections
and for mon1entum, and the letter R for resolvents and for S - I, where S is the
scattering operator. For the convenience of the reader we have included a Notation
Index (page 385). Constants are often denoted generically by c, but in some proofs
different constants are numbered as c1 , c2 , etc.
is a great pleasure to thank all those who helped me in one way or another during
the preparation of the lectures or of this volume. The problems in the text are mostly
due to Marius Mantoiu, Joachim Stubbe and Rafael Tiedra de Aldecoa; they assumed
the responsibility for the problem sessions with much competence and devotion. The
continual interest of my students and their constructive comments on earlier versions
of the text have influenced a considerable number of details. I received precious support from Philippe Jacquet, Andreas Malaspinas, Peter Wittwer and Luis Zuleta for
coming to grips with TEX -related difficulties. I thank Philippe Martin for proposing

the publication of my lectures, the referee for pointing out some errors and for useful
suggestions, and Fred Fenter from EPFL Pre,ss for advice and his very efficient management of the publishing process. Finally~' am indebted to the Physics Department
of the University of Geneva for its kind hospitality after my retirement.

Werner Amrein
Geneva, Switzerland
December 2008

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Contents
v

1

Hilbert Spaces
1.1 Definition and elementary properties
1.2 Vector-valued functions . . . . . . . .
1.3 Subsets and dual of a Hilbert space . .
1.4 Measures, integrals and LP spaces
Problems . . . . . . . . . . . . .

14
25

2

Linear Operators
2.1 The algebra B(H) . . . . . .

2.2 Projections and isometries
2.3 Compact operators . . . .
2.4 Unbounded operators . . . .
2.5 Multiplication operators . . . . . .
2.6 Resolvent and spectrum of an operator ..
2. 7 Perturbations of self-adjoint operators
... .
Appendix
Problems . . . . . . . . . . . . . . . . .

27
27
37
41
47
56
67
7
80
82

3

Symmetric Operators and their Extensions
3.1 The method of the Cay ley transform . .
3.2 Differential operators with constant coefficients
3.3 Schrodinger operators . . . . . . .
Appendix
Problems . . . . . . .


4

1
1
6
8

8

. . . .

Spectral Theory of Self-Adjoint Operators
4.1 Stieltjes measures . . . . . . . . . . . . .
4.2 Spectral measures . . . . . . . . . . . . .
4.3 Spectral parts of a self-adjoint operator .
4.4 The spectral theorem. The resolvent near the spectrum . . . .
....
Appendix: Proof of the Spectral Theorem
Problems . . . . . . . . . . . . . . . . . . . . . . . . .
0

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• • •

87
87
96
111
126

131
133

133
141
157
170
179
190


CONTENTS

Vlll

5

Evolution Groups and Scattering Theory
5.1 Evolution groups
5.2 Characterisation of the scattering states
..
5.3 Asymptotic condition. Wave operators .
5.4 Simple scattering systems. Scattering operator . .
5.5 Scattering operator and S- matrix .
5.6 Scattering cross sections
5.7 Bounds on scattering cross sections
5.8 Coulomb scattering
Problems
..
. ..


263
264
271
283
286
292
300
306
310
315

6 The Conjugate Operator Method
6.1 A simple example .
6.2 The method of differential inequalities .
6.3 The Mourre inequality
6.4 Application to Schrodinger operators ..
6.5 Relatively smooth operators
6.6 Higher order resolvent estimates
6.7 Some comtnutators
..
Appendix: Interpolation of operators
Problems

7 Further Topics in Scattering Theory
7.1 Asymptotic completeness .
7.2 Flux and scattering into cones
7.3 Time-independent scattering theory
7.4 The scattering matrix
7.5 Time delay

..

...

Appendix
Problems

References

I

..

193
193
204
213
218
226
235
243
252
260

317
317
324
343
352
364

374
379
381

Notation Index

385

Subject Index

389

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CHAPTER!

Hilbert Spaces

Hilbert space sets the stage for standard quantum theory: the pure states of a physical
system are identified with the unit rays of a Hilbert space 1{ and observables with selfadjoint operators acting in H. In this initial chapter we present the essential concepts
and prove the basic results concerning separable Hilbert spaces (Sections 1.1 - 1.3 ). In
Section 1.4 we then introduce £ 2 spaces, which are of special importance for quantum
mechanics. This requires some familiarity with measure theory, and we include a short
description of the necessary concepts from this theory.

1.1

Definition and elementary properties


Throughout this text a Hilbert space 1neans a co1nplex linear vector space,
equipped with a Hermitian scalar product, which is complete and admits a countable
basis. More precisely a (separable) Hilbert space His defined by the four postulates
(HI)- (H4) stated below:

(Hl) H is a linear vector space over the field CC o.f co1nplex nu1nbers:
With each couple {f, g} of elements of H there is associated another element of H,
denoted f + g, and with each couple {a, f}, a E CC, f E H, there is associated an element af ofH, and these associations have the following properties (where f, hE 1{
and a, {3 E CC):

f+g==g+f

f

a(f+g)==af+ag
a({3f)

+ (g + h) ==

(f +g) + h
(a+(3)f-af+(3f

== (a{3)f

If- f.

(1.1)
(1.2)

(1.3)


Furthermore there exists a unique eletnent 0 E H (called the zero vector) such that

Of== 0

O+f==f
1

Here 0 denotes the complex number a = 0.
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\If E H.

1

(1.4)


2

HILBERT SPACES

1i is equipped vvith a strictly positive scalar product 2 :
With each couple {j, g} of elements ofH there is associated a complex number (f, g),
and this association has the following properties 3 :

(q, f) == (.f, g)
\f, g + o:h) == \f, g)

+ a(f, h)


\J,J)>O

\lj,gEH

(1.5)

vaEC, \lj,g,hEH

(1.6)

except for f

(1.7)

==

0.

One then defines
(1.8)
(H3) 1i is complete:

Each Cauchy sequence in 1i has a lin1it in H. In other terms, if {fn}nEN is a sequence
elements of 1{ satisfying 4
li1n II fn - f m II
m,n-----+c:x:J

== o, s


(1.9)

then there exists an element f of 1i such that lim n-----+oo II f - fn II

== 0.

(H4) 1{ is separable:

1i has a countable orthonormal basis, i.e. there exists a sequence { e 1 , e 2 , e 3 , ... } of
elen1ents of 1{ such that
( 1.1 0)

and such that each element
' ...

f

of 1i can be expressed as a linear combination of

The dimension of a Hilbert space 1i is defined as the number of elements in an orthonormal basis. This number 1nay be finite or infinite, and it is independent of the chosen basis. In an infinite-dimensional space the postulate (H4) involves infinite linear
co1nbinations. These are interpreted as follows: if { e 1 , e 2 , e3 , ... } is an orthonormal
basis of 1i and f an arbitrary element of H, there E}Xists a sequence {a1 , a2 , a3 , ... } of
complex numbers such that f == Lkakek and llifll 2 == Lklakl 2 . The coefficient ak
this develop1nent off is given by O'k == (ek, f), and the series Lkakck converges
to f in the sense that II f - L~=l akck II ---+ 0 as N ---+ x.
elements of 1i are called vectors; they will be denoted by f, g or h (sometimes
e
vectors satisfying II ell == 1). The letters a and f3 will stand for complex numbers.
reader should be fatniliar with finite-dimensional Hilbert spaces from Linear Algebra. However many problems in quantu1n mechanics involve Hilbert spaces
infinite dimension. We shall see that, when suitably interpreted, some properties of

finite-ditnensional spaces have analogues in infinite-din1ensional spaces (for example
spectral properties of compact operators are similar to those of tnatrices ), while
2

Abo called "inner product" by sotne authors.

1

a denotes the complex conjugate of the cmnplex nutnber a.
{ 1, 2. ~3 .... } .

-+

N

5

i.e. for each

E

> 0 there exists a number N

N(s) > 0 such that

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llfn -

fm


II < E \In, m >

N.


DEFINITION AND ELEMENTARY PROPERTIES

3

other aspects of infinite-dimensional spaces do not appear in finite-dimensional ones
(for example in finite-dimensional spaces there is no distinction between weak and
strong convergence, and matrices do not have continuous spectrum).
Observe that, contrary to the convention adopted by mathematicians, the scalar
product is linear in the second entry and anti-linear in the first one. The non-negative
number I f I defined by ( 1.8) is called the norm of the vector This norm function
has the properties of a metric. The quantity II!- g\1 can be interpreted as the distance
fromftog, andonehas llafll == \cvlllfll, in particular 11-fll 11(-l)fll == llfll· The
scalar product can be expressed in terms of the norm of 1-i; this is the content of the
polarisation identity:

4\f,g) ==\if+ gl\ 2

-II!- gll 2 - ill!+ ig\\ 2 + illf- igll 2 ·

(1.11)

This equation is easily checked by using the definition of the norn1 and the linearity of
the scalar product [as in ( 1.16) below].
The following four inequalities are simple consequences of (H1) and (H2) 6 :


1\f,g)l < llfii·IIDII
Triangle inequality :
\if+ g\1 < 11!11 +\\gil
II!+ gll 2 < 2\1!11 2 + 2llg\\ 2
IIIJII-IIgll\ < llf- g\1.
PROOF. Iff== g, (1.12) is evident: (J,f) == llfl\ 2 . Iff#-

Schwarz inequality :

(1.12)
(1.13)

(1.14)
( 1.15)

g, assume for example

that g #- 0. Then, for any a E C:
0

<\If +ag\\ 2 == \f +ag,f

ag)

== 11!11 2

a(f,g) +a(g,f)

la\ 2 \lg\\ 2 .


(1.16)

a== -(f, g) /llgll 2 one obtains
o < 11!11 2 - : 21(.f,gW,
11 1
which implies (1.12) upon multiplication by llgll 2 . For (1.13) we take a== 1 in (1.16):
\If+ gl\ 2 == llf\1 2 + \J,g) + (g,f) + llg\\ 2
(1.17)
2
2
< llfl\ + I(J,g)l + \(g,f)\ + \\g\\
< llfl\ 2 + 2llfll·llgl\ + \\g\1 2 == (llfll + llg\1) 2 .

By taking

For (1.14) we observe that

o < llf- g\\ 2 == llfll 2 - \f,g)- (g,.f) + llgll 2 ,
hence

In ( 1.12), II f II · II g II denotes the product of the norms off and g. In the remainder of the text a product
of two norms will be written simply as 11·1111·11· With this convention, (1.12) will read l(f, g)J:::;; llfllllgJJ.
6

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4


HILBERT SPACES

Upon insertion
this inequality into ( 1.17) one obtains ( 1.14). Finally, to obtain
(1.15), supposeforexan1plethat Jlgll < 11!11; then, by using (1.13):
11!11- llgll

==

II!- g +gil

llgll

<

(II!- gil+ II gil)- llgll

==

o

II!- gil·

1.1 . 2. There are two types of convergence in a Hilbert space, called strong convergence and weak convergence, defined in terms of the norm and the scalar product
respectively:

Strong convergence: A sequence of vectors {fn}nEN in a Hilbert space His said to
converge strongly to a vector f E H if llfn - !II converges to 0 when n ~ oo. One
then writes s-limn_,.oo fn == f.
Weak convergence: A sequence of vectors {fn}nEN of H is said to converge weakly

to a vector f E H if, for each vector g E H, the sequence of complex numbers
{ \fn, g) }nEN converges to (f, g) when n ~ oo. One then writes w-limn---7 00 fn ==f.
The postulate (H3) is a Cauchy criterion for strong convergence: a strong Cauchy
sequence has a li1nit (in H). The Cauchy criterion is also true for weak convergence: if
{ fn} n EN is weakly Cauchy (i.e. such that, for each g E H, the sequence { (fn, g)} n EN
is a numerical Cauchy sequence), then there exists a vector f in H such that { f n}
converges weakly to f. By using ( 1. 7) it is easy to prove the uniqueness of the limit of
Cauchy sequences (see Proposition 1.3).

Proposition 1.1. (a) Let {fn }nEN be a sequence of vectors in H. Then
s -liln Jn
n
oo

== f

¢===:;>

---7

(b) ffs-lin171---7oo fn

w - lim f n
n
oo

== f

and


li1n II fn II
oo

n

---7

== II f

II·

---7

== f and s-lillln---7oo gn == g, then limn---7oo \fn, gn) == (J, g).

PROOF. (a) Assume that s-limn---7oo fn ==f. By using (1.12) one obtains that

1\fn,g)- (J,g)l == 1\fn- f,g)l < llfn- fllllgll ~ 0

as n ~ oo,

~~

so w-limn__, 00 .fn = f. On the other hand, by using (1.15\ one obtains lllfn 11-11.!111 <
II fn - !II ---+ 0 as n ---+ oo, hence limn---7oo II fnll == II fll·
2
obtain the implication~ we observe that II fn- !11 2 == II fn 11 2 +II !11 - (fn, f)\f~fn)·
w-linln---7oofn == f andlimn---7oo llfnll == 11!11, the expression on the right2
2
2

hand side converges to 11!11 + 11!11 2 - llfll - 11!11 == 0, so that s-lin1n---7oo fn ==f.
(b) This follows from (1.12) and the result of (a):

.gn)- (J.g)l == 1\fn,gn- g)

< llfniiiiYn

(fn- f,g)l

< 1\fn,gn- g)l

gil+ llfn- fllllgll ~ II Jll · 0

+ 1\fn- f,g)l

0 · llgll == 0.

D

Example 1.2. Let { e 77 } nEN be an infinite orthonormal sequence in a Hilbert space
H of infinite dimension, i.e. satisfying (e 1 , ek) == 61k. Then w-limn---7oo en == 0,
because for any vector gin H:
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DEFINITION AND ELEMENTARY PROPERTIES

5

00


( 1.18)
n=l
consequently limn-'roo (en, g) == 0 [the sequence {en} can be considered to be part of
an orthonormal basis of H, and for such a basis one has equality in ( 1.18) as explained
afterEq. (1.10)]. Thus:
Every il1finite orthonormal sequence {en} converges weakly to the zero vector 0.

On the other hand such a sequence cannot be strongly convergent (if one assumes
that s-limn-'roo en== f, then w-linln-'roo en== f by Proposition 1.1(a), and by the
uniqueness of a weak limit, one must then have f == 0, i.e. s -li1n 11 -'r()() e 11 == 0; this is
impossible since lien- Oil == lien II == 1 for each n). This example shows in particular
that in an infinite-dimensional Hilbert space the notion of weak convergence is really
weaker than that of strong convergence.

Proposition 1.3. The limit of a strong or weak Cauchy sequence is unique.
PROOF. If {fn} is a strong Cauchy sequence, it is also weakly Cauchy [see Proposition 1.1 (a)]; thus it is enough to consider weak convergence. So let us assume that
w-lilTI 11 -'roo fn == h1 and w-lilnn-'roo fn == h2. Then one has for each g E H:

(g, h1 - h2) == (g, h1) - (g, h2) == lim (g, fn) - lim (g, fn)
n-'roo
n-'roo
== n-'roo
lirn [(g, fn) - (g, fn)] == 0.
Upon taking g == h1 - h2, one obtains that II h1 - h2ll 2
h1 - h2 == 0, i.e. h1 == h2.

==

0. By (1.7) one then has

D

REMARK. If {fn}nEN is a strong Cauchy sequence, then this sequence is bounded in
H, i.e. supnEN II f n II < 00. This follows immediately from Proposition 1.1 (a) (because
llfnll ---+ 11!11). The same conclusion is true if one assumes only that {fn} is weakly
Cauchy, but the proof is longer. This fact is often useful in the theory of Hilbert spaces,
and we shall see in Chapter 2 other results of this type. All these results can be deduced
from a general theorem (the Uniform Boundedness Principle) which we state below
for completeness (the proof can be found in most textbooks, see for example [K]).

Uniform Roundedness Principle: Let A be a set and, for each element A of A, let
..: H ---+ [0, oo) be a continuous mapping 7 such that .. (f
g) < .. (f) + .. (g)
for all J, g E H. If for each fixed vector g the family { .. (g)} >-.EA is bounded (i.e.
.. (g) < M \1 A E A, where M < oo is a constant depending only on g), then the
family { cp >-.} is uniformly bounded on the unit ball of H, in other terms there exists a
constant c < oo such that cp;>..(h) < c for all hE H satisfying llhll < 1 and all A EA.
To deduce for example the boundedness of a weakly convergent sequence { fn} nEN,
take A == N and cp >-.(g) cpn (g) == I(fn, g) I· For each g E H, the numerical sequence
{I (fn, g) I} nEN is bounded because (fn, g) converges to a finite limit; then one uses
the fact that llfn II== suphEH,Jihll=l 1\fn, h) I [see Eq. (2.6)] to arrive at the inequality
II fn II == suphE'H.JJhlJ=l cpn (h) ~ c for all n.
7

i.e. cpA (fn)

---+

cpA (f) when

fn

---+

f

strongly.
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HILBERT SPACES

6

Vector-valued functions
Let A be a set and H a Hilbert space. A vector-valued function on A is a mapping
f: A ~ H. It associates a vector in H with each ele1nent of A. Thus a sequence of
vectors {fn}nEN can be viewed as a function f: N ~ H by setting f(n) == fn for
n EN. The 1nost important case in our context is that where A is a (finite or infinite)
interval ~l; we write f (s) for the value of a function f: J ~ H at the point s E J. So
for each s E J, f (s) is a vector in H. Iff (s) depends continuously on s, the family
{!( s) }sEJ describes a curve in H.
As for nu1nerical functions, one can consider the notions of continuity or differentiability and define integrals of such functions. The derivative at a point of J and the
integral of a vector-valued function will be vectors in H. Since derivatives and integrals are limits, these notions may be defined in the strong sense or in the weak sense
(for example a function f: J ~ H may be weakly differentiable but not strongly
differentiable). Similarly one can define integrals in the sense of Riemann or in a
In ore general sense, and one can integrate with respect to different measures on J (the
concept of a measure will be discussed in Section 1.4). We restrict ourselves here to
definitions in the strong topology of H, and we consider only integrals in the sense of
Riemann (and with respect to Lebesgue measure).

~His

A function f: J

strongly continuous if one has for each t E .1:

lirn
s-d,sEJ

I ! (s) - ! (t) I

==

o.

( 1.19)

A vector-valued function f on an open interval J is strongly differentiable at the point
s E tl if there is a vector g in H such that:

~-To II* [.f(s + T)- .f(s)] -

g

II= 0.

(1.20)

We shall write g == f' (s). More generally, f is strongly differentiable in J iff is differentiable at each point s E J, i.e. if there exists a function : J ~ H such that

~-=!~ I *[f (s + T) - .f (s) J - f' (s) I =0\ vs E J.
The function

f'

is called the (strong) derivative

f S)
1

(

== dd

S

f (S)

=

( 1.21)

off, and one also writes

S- lim T- l [ f (S + T) - f (S) J ·

(1.22)

T-+0

To define the Riemann integral of a vector-valued function, one proceeds in analogy
with the construction for numerical functions. Let J == (a, b] be a finite interval. A
collection of real numbers II== {so, s1, ... , sN; u1, ... , UN}, with a== s0 < s1 <
· · · < s N == band with Uk E ( s k- 1 , s k], will be called a partition of J (Figure 1.1). We
set IITI == maxk=1, ... ,N isk- Sk-11·
Iff: J ~ H, let
N

L:(IT,f) ==

L (sk- sk-1)f(uk)·
k=1

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(1.23)


VECTOR-VALUED FUNCTIONS

7

This is a finite linear combination of vectors in H, hence it defines a vector I:(IT,f)
in H. One chooses a sequence {I1r }rEN of partitions of J such that lin17'-""0C IITr I== 0
and defines

/

.l

. f( s) ds
c

J

6

f( s) ds = s-lim I::(Il,.,f)

(1.24)

r~~

a

exists and is independent of the sequence {I1r}.

o

(

Bo

8 3 . . . . . . . . . . . . . . . . . . . . . 8 N-l

>
)(

8N

b

I><]

11,~3 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 1L N

Figure 1.1 Partition of an interval (o, b].

The integral of a vector-valued function on a finite open interval (a, b) or on a
finite closed interval [a, b] is defined similarly by suitably adjusting the definition of
a partition of J: if J == (a, b), then one must assume that UN
b, and if tl == [a, b],
then u 1 == a is also admissible.
If J is an infinite interval, one first defines the integral for each finite subinterval
and then takes a sequence of finite subintervals converging to J in an appropriate
way. For exa1nple, J == (a, oo) for some finite a, one defines (if the limit exists)
00

.l .f(
a

s) ds

=

s- lim / b .f (8) d s.
b----7~ .fa

( 1.25)

We state below some simple properties of these integrals and sufficient conditions for
their existence. The proofs are analogous to those for numerical (real or co1nplex)
Riemann integrals; it suffices to replace in the latter proofs the absolute value If (s) I
by the norm llf(s)ll (details can be found for example in Section 4.4 of [AJS]).
Proposition 1 . 4 . Let (a, b] and (b, c] be finite or infinite intervals and suppose that all
integrals below exist. Then
( 1.26)
( 1.27)
( 1.28)
(i) Assume that a and bare finite. Then (1.28) is obtained by using Proposition 1.1 (a), the triangle inequality (1.13) and the definition of the numerical Riemann
integral:

PROOF.

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8

HILBERT SPACES

(ii) If for exa1nple a is finite and b
above that

== oo, one finds from Proposition

1.1 and (i)

Proposition 1.5. (a) ff [a, b] is a finite closed interval and f: [a, b] ---+ H is strongly

continuous, then ,[abf ( s) ds exists.
(b) ff a and b are arbitrary (a < b), f: (a, b) ---+ H is strongly continuous and
fab llf(s)llds < oo, then fabf(s)ds exists.
(c) Iff is strongly differentiable on (a, b) and its derivative f' is strongly continuous
and integrable on [a, b], then

1bf'(s)ds

=

J(b)- J(a).

( 1.29)

Notice that, iff is strongly continuous, then llf(s)ll depends continuously on s by
Proposition 1.1 (a), so that ,[ab I f (s) I ds can be defined as a numerical Riemann integral.

Subsets and dual of a Hilbert space
We consider here certain types of subsets of a Hilbert space H. A subset D of
H is dense in H if, given any vector f E H and E > 0, there exists an element g of
D such that II!- gJI < c. Equivalently, Dis dense in H if, given any vector f E H,
is a sequence {fn }nEN in D such that liinn-+oo II/- fnll == 0. An example of
a dense set is given by the collection
all finite linear combinations of vectors of
an orthononnal basis {en} of H. In this case D is the set of all vectors of the form
, with ak E C and N == 1, 2, 3, ... (N < oo); if one assumes in addition
1 CYk
real part and the imaginary part each a k are rational nu1nbers, one obtains
a dense subset of H that is countable.
linear manifold£ is a linear subset of H, i.e. such that f, g

and c1: E C ====?
f cxg E £ (in other words each finite linear combination of elements of £ also
belongs to£). An exa1nple is the already indicated set of all finite linear combinations
of
of an orthonormal basis {en} of H; more generally, if N is an arbitrary
subset
H, the set
all finite linear combinations
vectors belonging to N is
a linear 1nanifold, called the linear manifold spanned by N. If the linear manifold
spanned by a subset N is dense in H, one says that N is a total set in H or a total
1-l. An exa1nple a non-total subset is given by the vectors e 2 , e 4 , e 6 , ... of
an orthononnal basis {en}.
linear 1nanifold £ in H is clearly a complex linear vector space equipped with a
product (induced by the scalar product in H, i.e. if J, g E £, then their scalar
£ is sin1ply their scalar product (/,g) in H). In general £ will not
product (f. g) E

d

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i


SUBSETS AND DUAL OF

9

HILBERT SPACE

be complete in the norm 11·11, i.e. the limit of a Cauchy sequence {fn}, with fn E £,will
not belong to£ but only to H. If£ is cornplete [i.e. if£ satisfies all of the postulates
(Hl)- (H4)], then£ will be called a subspace of H. In what follows we use the syrnbol
M for subspaces.
A linear manifold£ determines uniquely a subspace£, called the closure of£: a
vector f in H belongs to£ if and only if there exists a sequence {fn }nEN in £ that
converges strongly to f as n ---+ oo. If£ is dense H, then£ == H, otherwise£ is a
subspace of H that is strictly smaller than H.
More generally, an arbitrary subset N of H defines uniquely a subspace M.!V,
called the subspace spanned by N: MN is the smallest subspace containing N, in
other terms the closure £ of the linear manifold £ spanned by N.
EXAMPLES OF SUBSPACES
(a) A subspace of dimension 0 is {0}, consisting of only the zero vector.
0 determines a one-dimensional subspace, the set { af
E C},
(b) Each vector f
i.e. all multiples of the vector f.
(c) A subspace of infinite dimension of an infinite-dimensional Hilbert space H is
obtained by taking all (finite and infinite) linear combinations belonging to H of the
vectors e2, , e6, ... of an orthonormal basis { en}nEN of H.

\a

REMARK. The terminology concerning subspaces is not uniform; sotne authors say
"subspace" for a linear manifold (they have in mind a subspace of the vector space H)
and "closed subspace" for what we call a subspace.
The scalar product leads to the notion of orthogonality in H. Two vectors f and g
of H are said to be orthogonal if (j, g) == 0, and one then writes f _L g. A sequence
{!1 , f 2, j 3 , ... } of vectors of His called orthonormal if (f~, fk) == 5.Jk·

If N is a subset of H, one defines its orthogonal comple1nent N _L as the set all
vectors in H that are orthogonal to each vector inN:
N_L=={JEH\(J,g)-o vgEN}.

(1.30)

We write f _L N iff E N _L. The set N _L is a subspace of H (Problern 1.7).
the subspace spanned by N, one has

MN is

( 1.31)

Indeed: (i) the implication <¢== is trivial. (ii) For the implication =? we first observe
that f _L N implies f _L £, where £ is the linear manifold spanned by N (because
each g E £ is of the form g == 'I:~=l akgk with .9k E N for some N < CXJ, hence
(f, g) == 'I:~=l ak (f, gk) == 0). Finally f
£ implies that f
£
MN (because
each element g of the subspace£ is given as g == s-linln-+oo .9n with .9n E £,hence
(.f,g) == lirnn-+oo (f,gn) == linln-+oo 0 == 0).

Proposition 1.6. A subset
f == 0.

N o.f H is total in H
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~f and only

iff

N inlplies that


10

HILBERT SPACES

PROOF.
N is total in H and f N, then f is orthogonal to H by ( 1.31 ). In particular f is then orthogonal to itself, i.e. (f, f)== llfll 2 == 0, hence f == 0 by (1.7).
Conversely, suppose that f N implies that f == 0. By virtue of (1.31) this means
that the orthogonal co1nplement of the subspace MN spanned by N is {0}. Hence
Mjv == {O}_L == H (here one applies the Projection Theorem given below).
D
The next proposition generalises a well-known property of finite-dimensional vector spaces equipped with a scalar product. This is a fundamental result in the theory of
Hilbert spaces, and it also holds in non-separable Hilbert spaces [the postulate (H4)
not used in its proof].

Proposition 1.7 (Projection Theorem). Let M be a subspace of H, M_L its orthogonal cornplement. Then each vector f qf H admits a unique decomposition into a
con1ponent in M and a conzponent in M _L:
( 1.32)
PROOF. We fix a vector f E H, and we denote by d the distance from f to M, i.e.
d == infgE.A1 I f gjl. We shall use the following identity: if h 1 and h2 are vectors in
then, by Eq. ( .17) with f == h 1 and g == -::r-h2:
( 1.33)

== I f - go II·
~~~ Choose a sequence {gk}kEN in M such that 1in1k----+oo ll.f- gkli == d. Let

us take h 1 - (f- g.7) /2 and h 2 == (f - gk) /2 in (1.33). We obtain
We first show that there is a unique vector g0 E M such that d

Since g1

+ 9k EM, we have II f

~ llrh

gk) /211 2 > d 2 . Hence

- (g.J

gkll <~III- gJII +~III- gkll
2

2

2

2

-

d -'to

asj, k -'too.

{gk} is a Cauchy sequence H. We denote its litnit by 9-D· Since M is a subspace, we have go EM. Consequently I f- go II > d. On the oth~r hand

llf- 9oll have thus shown that II f

ll9k- go II---+ d

- go II -

0

== d ask---+ oo.

d.

Uniqueness. Let us assume that there exist two vectors gg) and g~2 ) in M such that

/If- g~1 )11 == II/- g62)11
1

==d. By taking h1
( 1.33) one obtains as in (i) that

41 II 9o(I)

2 2
) 11

9o(

II f
-< _!_

2 .

- .Jo
c (I)

==
2

11

. go(1) -- 9o(2) .
Th us II rJo(I ) - go( 2) II - 0, l.e.
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- g~l))/2 and h2 == (f-

g62))/2 in


SUBSETS AND DlJAL OF A HILBERT SPACE

1

(ii)
prove (1.32) we set / 1 == g 0 , where g 0 is the unique vector in M associated
with f according to (i). Then f 2 == f - g0 , and we have to show that / 2 E Mj_, in
other terms that (h, / 2 ) == 0 for each h E M. So let h #- 0 be a fixed vector in M
and set a== (h, /2). Let g == /1 + ah/llhll 2. Clearly gEM, hence II!- .c;ll > d and
consequently

This implies that lal 2 < 0, hence that a == 0.
(iii) Finally let us check the uniqueness of the decomposition ( 1.32). Assume that
g1, g2 E M and h1, h2 E M _i are such that g1 + h1 == 92
h2. We must show that
g1 == .92 and h1 == h2. Now
0

== IIOII 2 == 11.91 + hl- (g2 + h2)11 2
== Jl(gl- .92) + (hl- h2)11 2 == 11.91-.9211 2

because (g1 - g2, h1 - h2)
h1 == h2 by (1.7).

==

0. Thus Jlgl - g21i

2
Jlht- h211 .

== II h1 - h2ll ==

0, i.e. 91

== .92

and
D

The projection theorem shows that H may be viewed as the orthogonal (direct) sum

of two mutually orthogonal subspaces M and M _i. We recall that the orthogonal sum
H H1 E8H2 · · ·ffi'HN of N Hilbert spaces H 1, ... , HN is the Hilbert space forn1ed
by all N-tuples {/1 , ... ,
}, where
E Hk, with the following scalar product:
( 1.34)

Each Hk is a subspace of H, and if j #- k then the spaces H.7 and Hk, considered as
subspaces of H, are mutually orthogonal.
Occasionally we shall also use orthogonal sums of an infinite number
Hilbert
spaces. The orthogonal sum H ==
1Hk of an infinite sequence {H1, H2, .. .} of
Hilbert spaces is formed by all infinite sequences {/1 , f 2, ... } , with fk E Hk and such
that ~Z: 1 11 ll~k < oo, the scalar product in H being defined as in (1.34) but with
N == oo. Again each Hk is a subspace of H (the set of all sequences {/1 , , ... } with
f 7 - 0 for all j k), and the subspaces H.7 and Hk are mutually orthogonal if j #- k.
As an example, suppose that each Hk is a one-dimensional space, i.e. Hk == C for
each k. If N is finite, the orthogonal sum H == H 1 E9 H 2 E9 · · · E91iN is (isomorphic
to) the space CN. If N == oo one obtains the space denoted by I! 2 , the Hilbert space of
all infinite sequences { a1, a2, ... } of complex numbers satisfying ~ZO= 1 1ak 12 < oo,
with scalar product

EBZ:

00

\{al,a2, ... },{Pl,P2,···}) == LakPk·
k=1

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(1.35)


12

HILBERT SPACES

Each (separable) infinite-dimensional Hilbert space His isomorphic toR 2. Indeed H
may be identified in a natural way with f 2 by choosing in H an orthonormal basis
{en} and by identifying the vector f == ~kakek (see Section 1.1) with the element
off 2 given by the sequence { a 1, a 2, ... } . In the terminology that will be introduced
in §2.2.2, the correspondence H 3 f t-7 { a 1, a 2 , ... } defines a unitary operator from
H to f 2 .

1.3.2. We end this section with a characterisation of the dual of a Hilbert space. The

dual H* of H is by definition the set of all bounded linear functionals on H, i.e. the
set of all mappings
cp(f

---+

C having the following two properties:

ag)

==

cp(f)

Jcp(f)J

<

cJJfll

+ acp(g)

(linearity)
(boundedness)

\If E H,

where cis a constant (independent of f). The space H* is a complex linear vector
space (for example the sum of two bounded linear functionals is again a bounded
linear functional), and H* is also normed with the norm

(1.36)
Each vector g of H detennines an element

(f?g (f) ==

(where f varies over H).

(g' f)

(1.37)

"

Indeed, the scalar product (for fix~d g) is linear in f, and [use (1.12)]

In fact one has

(1.38)
because the supremum is attained for f == g: (g, g) /Jig II== jjgjj 2 /II gil== llgJJ.
We have thus shown that each vector of H determines an element of the dual space
H*. It is an important fact that the converse is also true (so that one may identify H*
and H):

Proposition 1.8 (Riesz Lemma). Let H be a Hilbert space and cp a bounded linear
functional on H. Then there exists a unique vector g in H such that
cp(f)
In particular:
PROOF. (i) Define

llcpiiH*

==

vf

(g, f)

== jjgjJH

E

H.

(1.39)
(1.40)

jjgjj.

M to be the set of all vectors f E H for which cp(f) == 0. Let us

show that M is a subspace:
( 1) f1· f2 EM and()! E C ====? cp(fi

af2) == cp(fi)
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+ acp(f2) == 0

0

== 0,


SUBSETS AND DUAL OF A HILBERT SPACE

13

(2) if {fn} is a sequence of vectors in M such that limm,n-r(X) llfnL- fnll == 0, then
the sequence {fn} has a li1nit fin 7-i, and we must show that this vector f belongs to
M (completeness of M), i.e. that cp(f) == 0. Now

because cp(fn) == 0. As llf- fnll ~ 0 when rt ~ oo and cp is bounded, one must
have cp(f) == 0.
(ii) If cp
0 we take 9 == 0; indeed one then has 0 == cp(f) == (0, f) for each
f E 7-i. If cp ¥::- 0 there is a vector h in 1i such that cp(h) i- 0. By the Projection
Theorem we can write h == h 1 + h 2 with h 1 E M and h 2 E M _L. Then
cp( h2)

== cp( h

h1)

== cp( h) - cp(h1) == cp( h)
~

i- 0,

0

in particular h 2 i- 0 (because cp(O) == 0).
Iff is an arbitrary vector in 7-i, let us consider the vector f- [cp(f)/cp(h 2)] h 2 . We
then have

hence the vector f- [cp(f)/cp(h 2)] h 2 belongs toM. Since h 2

j_

M we obtain

Upon setting 9 == [ cp( h2) /II h2ll 2] h2 (which defines a vector in 1i because I h2ll i- 0),

one obtains ( 1.39).
(iii) We finally show that 9 is unique. If 9 1, 92 are two vectors such that cp(f) ==
(91, f) == (92, f) for all f E 7-i, then (91 -92, f) == 0 for all f E 7-i. Hence 91 - 92 == 0
by Proposition 1.6.
D
Let us add that the identification of 7-i* with 1i is not linear but anti-linear: if cp 1
and C;J2 are elements of 7-i* such that cp 1(f) == (9 1, f) and cp 2 (f) == (9 2 , f) for all
f E 7-i, then

hence the vector in 1i representing the linear functional cp 1
a92·
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+ (~cp 2

is the vector 9 1 +


14

HILBERT SPACES

1.4 Measures, integrals and LP spaces
rigourous treatlnent of the theory of L P spaces requires familiarity with measure
theory and integration in the sense of Lebesgue. We shall here present a description
of these spaces and of so1ne aspects of measure and integration theory, without giving
the proofs. The co1nplete theory is treated in numerous books.

1.4. 1. Consider a set 0 and a collection A of subsets of 0 having the structure of a
CT-algebra, i.e. satisfying: ( 1) A contains the empty set 0 and 0 itself, (2) if V E A,

then its complementary set 0\ V also belongs to A, (3) A is stable under the operation
taking countable unions in the sense of set theory: if Vk E A for k == 1, 2, 3, ... , then

UkVk E A. If these conditions are satisfied, then A

is also stable under the operation
of taking countable intersections, and the couple (0, A) is called a measurable space.
We use the letter x for the elements of 0 (the "points"). A measure m on (0, A) is
a mapping from A to [0, oo] that is a--additive, i.e. such that rn(Uk Vk) == I:km~(Vk)
for each countable family {Vk} of disjoint elements of A (i.e. satisfying ~in Vk == 0
if j i- k ), and with rn( 0) == 0. A measurable space (0, A) together with a measure m~
defined on it is called a measure space, denoted by (0, A, rn ).
We see that a measure rn on (0, A) assigns a "weight" (which may be infinite)
to each subset V of 0 belonging to A, and this assignment is additive (the weight
of a finite or countable union of disjoint subsets of 0 is the sum of the weights of
these subsets). A set V of 1neasure zero will be called a null set with respect to m. If
rn( 0) < oo, then 1n is called a finite measure. If 0 can be expressed as the union of
a countable collection of elements of A each of which is of finite measure, then rn is
called aCT-finite measure. In particular every finite measure is a--finite. An example
of a a--finite n1easure that is not finite is given by~ Lebesgue measure on JR. (see
§ .4.2). A finite measure with rn( 0) == 1 is a probability measure; in applications
the set 0 then represents the possible outcomes of an experiment, the elements of the
a--algebra A correspond to the events of interest in the experiment, and the measure
rn (V) is the probability that the event V will occur (an example in quantum mechanics
will be discussed on page 168).
A function cp: 0 ~ JR. assigns a real number cp(x) to each point x of 0. Given a
a-algebra A of subsets of 0, such a function cp is said to be a measurable function
if. for each interval J in IR, the set cp- 1 (.J) :== {x E 0 I cp(x) E J} belongs to the a-algebra A. It is seen that measurability is not simply a property of the function cp but
depends very much on the a- -algebra A. cp may be measurable with respect to certain
a--algebras and non-measurable with respect to other a--algebras of subsets of 0.

The important fact for integration theory in the sense of Lebesgue is that each
n1easurable function is the limit of a sequence of simple measurable functions (one
could take this characterisation as the definition of a measurable function). A simple
function is a function of the form cp == I:~= I akXv" with ak E JR. and N < oo,
where Xv,", is the characteristic function of the subset Vk of 0; a simple function is

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MEASURES, INTEGRALS AND L P SPACES

15

measurable if
E A for k == 1, ... , N. We recall that the characteristic junction
of a set V is defined as
XV (X) == 1

if X E V,

XV (X) == 0

if X Et V.

xv

( 1.41)

Thus, if cp a measurable function, there exists a sequence { cpn} of measurable si1nple
functions

that cp(x) ==limn-too cpn (x) for each x E 0. Furthermore, if cp > 0, the
sequence {cpn} can be chosen non-decreasing, i.e. such that 0 < cprz (x) :::; cprn (x) <
cp(x)
each x E 0 and m > n.
If cp is a measurable simple function, say cp == ~~= 1 akXv", then the integral of cp
with respect to the 1neasure m on (0, A) is defined to be the real number

1

N

cp(x)m(dx)

=

0

Lo,.m(Vk)·

( 1.42)

k=l

This has a sense provided that the sets Vk are such that nt(Vk) < CXJ. If this is the case,
the function cp == ~~=l ak Xvk is called a simple m-integrable function.
If cp is a general measurable function (not necessarily simple), one can define its
integral by approximating cp by a sequence of simple functions { 4?n}. One considers
first measurable functions cp > 0 and defines

.!.

cp(.T)m(dx)

0

=

lim
n-too

1

cpn(x)nl(d:r),

(1.43)

0

cp( 1;) for all
where { 4Jn} is such that 0 < cpn (x) < cp( x) and lirr1 n -too cpn (x)
x E 0. This makes sense if each of the simple approximating functions 4?n is rnintegrable and if the limit in (1.43) is finite. If cp is not> 0, one applies the preceding
definition to its positive and its negative part, i.e. one decomposes cp into cp with
> 0 and cp_ > 0; the integral of
and that of 4?- are defined as in (1.43),
and cp is called m-integrable if each of these two integrals is finite. Then one sets

r cp(x)m(d:r:)

Jo

=

1o

(.T )m( dx) -

./~ cp_ (:r)m (d:r ).

(1

If cp: 0 ~ CC is a complex function, it is rn-integrable if its real part and its imaginary part are rn-integrable in the sense of the preceding definition. The integral of cp
is obtained by integrating separately its real part and its itnaginary part.
The Lebesgue type integral introduced above generalises the (proper) Riemann integral on the real line. As an illustration we shall co1npare in §1.4.2 below the Lebesgue integral on IR with the Rietnann integral for continuous functions. For measurable functions without continuity properties, only the Lebesgue integral has a tneanIng.
In the situations considered in this text, the underlying set 0 will be a subset of IRn
(n) == 1, 2, 3, ... ) , often a finite open interval (a, b) or a semi-infinite interval (a. CXJ),
or IRn itself. In each of these situations the a--algebra A will be the Borel a--algebra
or the Lebesgue a--algebra of the considered set 0. Mostly the measure rn will be the
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HILBERT SPACES

16

Lebesgue 111easure m( dx) == dx [m( dx) == dnx if n > 2]. In the next subsection we
explain these tenns in somewhat more detail.

1.4.2. B is an arbitrary collection of subsets of a set 0, there is a smallest a-algebra
A containing B (i.e. such that, if A' is a a-algebra with B C A', then A C A'). This
1ninimal a-algebra A, called the a-algebra generated by B, is simply the intersection

of all a-algebras containing B (this intersection is not empty because the collection of
all subsets of 0 is a a-algebra). In certain situations a measure m, on A is completely
detern1ined if it is given on !3. An important example is the Borel-Lebesgue measure
on the real line which we shall now describe.
So let us consider the case 0 == IR. We take for !3 the set of all half-open intervals
(a, b] with -oo < a < b < +oo. The a-algebra generated by the collection of all
these intervals is called the Borel a-algebra of IR; we denote it by AB, and its eleInents will be called Borel sets. 8 Let us take for the measure of an interval (a, b] its
b - a. The thus defined mapping m: B ~
length R, i.e. 112,( (a, b]) == R( (a, b])
[0, oo) has an extension to a 1neasure on (IR, AB ), often called the Borel measure on
IR. This 1neasure associates with each Borel set of IR a non-negative number (which
can be infinite) in a a-additive way, and the number associated with an interval is its
length. The length of a general Borel set V is obtained in the following manner: one
considers covers of V of the fonn V C Uk Jk, where { Jk} kEN is a countable collection
half-open intervals [Jk == (ak, bk], ak < bk], and one then defines

R(V) ==infLR(Jk)

infL(bk -ak),

k

(1.45)

k

where the infi1num is over all covers of V of the form specified above. This length
function has
properties of a measure on (IR, AB ), viz) the Borel measure mB on
the real line: 1nB (V) == R(V) for V E AB·

We 111ention some examples of null sets with respect to Borel measure (Borel sets
length zero). The simplest example is a single point { x 0 }. This set is the intersection of all intervals (x 0 - k- 1 , x 0 ], k == 1, 2, 3, .... The length of such an interval is
7n~ B ( (;:r 0 - k- 1 , x 0 ]) == 1/ k; since { x 0 } is contained in each of these intervals, its measure must be less than 1/ k for each k == 1, 2, 3, ... , hence equal to zero. 9 A second example is that of a countable set { x 1 , x 2 , x 3 , ... } ; as m B is a -additive and the measure
of each {xk} is zero, one has mB({x1,x2,x3, ... }) == LZ: 1 n~B({xk}) == 0. There
also exist Borel null sets that are not countable, for example the Cantor set is a subset of the interval [0, 1] obtained by removing from [0, 1] a countable union
8 The

smne a-algebra is obtained by starting with the set of all open intervab or with that of all closed
intervals. Thus AB contain~ all open, closed and half-open intervals. In fact each open and each closed
subset of IR is a Borel set (this is a consequence of the fact that each open subset of IR is the union of
countably many disjoint open intervals), but there is no simple characterisation of individual Borel sets
directly in tenns of unions and intersections of open and closed sets. The use of intervals of the forn1 (a, b]
n1ay be ju~tified by the fact that ~uch intervals fit together in a nice way, and this will be crucial for the
definition of the n1ore general Stiel~jes n1easures (see Ren1ark 4.6).
9
A basic property of a measure rn is that, if V, W E A are such that V ~ W, then m (V) :s; m(W),
which is a ~traightforward consequence of the fact that a 111easure is additive.
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