Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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4.8
PROBLEMS
141
Additional Reading
1. Alder, S. Linear Algebra Done Right, Springer-Verlag, 1996. Concise but
useful discussion
of
real and complex spectral theory.
2. DeVito,
C. Functional Analysis and Linear Operator Theory, Addison-
Wesley, 1990. Has a gooddiscussion
of
spectral theory for finite and infinite
dimensions.
3. Halmos,P. FiniteDimensionalVectorSpaces, 2nd ed., Van Nostrand, 1958.
Comprehensive treatment
of
real and complex spectral theory for operators
on
finite
dimensional vectorspaces.
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Part II _
Infinite-Dimensional Vector
Spaces
5 _
Hilbert Spaces
The basic concepts of finite-dimensional vector spaces introducedin Chapter I can
readily be generalized to infinite dimensions. The definition
of
a vector space and
concepts of
linear
combination,
linear
independence,
basis,
subspace,
span,
andso
forth all carry over toinfinite dimensions. However, one thing is crucially different
in the new situation, and this difference makes the study of infinite-dimensional
vector
spaces
both
richer
and
more
nontrivial:
In afinite-dimensional vectorspace
we
dealt
with
finite
sums;
in
infinite
dimensions we
encounter
infinite
sums.
Thus,
wehaveto
investigate
the
convergence
of such
sums.
5.1 The Questionof Convergence
The intuitive notion
of
convergence acquired in calculus makes use
of
the idea
of
closeness. This, in tum, requires the notion
of
distance.' We considered such a
notion in Chapter I in the context of a norm, and saw that the inner product had
anassociated
norm.
However,
itispossibleto
introduce
a
norm
onavector
space
without
an
inner
product.
One such norm, applicable to
en
and
JR
n,
was
where p is an integer. The "natural" norm, i.e., that induced on
en
(or
JRn)
by
the usual inner product, corresponds to
p = 2. The distance between two points
1It is possible to
introduce
the idea of closeness
abstractly,
without
resort
to thenotionof distance,as is done intopology.
However,
distance,as appliedin vectorspaces,is as
abstract
as we wantto get.
146 5. HILBERT
SPACES
Closeness
isa
relative
concept!
Cauchy
sequence
defined
complete
vector
space
defined
depends on the particntarnorm nsed.
For
example, considerthe "point" (or vector)
Ib) =
(0.1,0.1,
...
,0.1)
in a 1000-dimensional space (n = 1000).
One
can easily
check that the distance
of
this vector from the origin varies considerably with p:
IIblil
= 100,
IIbll2
= 3.16, lib
II
10 = 0.2. This variation
may
give the impression
thatthere is no snchthing as "closeness", and it all depends on how
one
defines the
norm. Thisisnot true, becanseclosenessis arelativeconcept: One always
compares
distances. A norm with large p shrinks all distances
of
a space, and a
norm
with
small
p stretches them. Thus, although it is impossible (and meaningless) to say
that
"Ia) is close to Ib)" because
of
the dependence
of
distance on
p,
one can
always say
"Ia) is closer to Ib) than [c) is to Id)," regardless
of
the value
of
p,
Now that we have a way
of
telling whether vectors are close together or far
apart, we can talk about limits and the convergence
of
sequences
of
vectors. Let
us begin by recalling the definition
of
a Cauchy sequence
5.1.1. Definition. An infinite sequence
of
vectors
{lai)}~l
in a normed linear
space
V is called a Cauchy sequence if
1im;-->oo
lIa;
-
aj
II
=
O.
j-HX)
A convergent sequence is necessarily Cauchy. This can be shown using the
triangle inequality (see Problem 5.2). However, there may be Cauchy sequences
in a given vector space that do
not
converge to any vector in that space (see the
example below). Snch a convergence requires additional properties
of
a vector
space summarized in the following deliuition.
5.1.2. Definition. A complete vector space Vis a normed linear space for which
every Cauchy sequence
of
vectors in V has a limit vector in V. In other words,
if {Iai)
}~l
is a Cauchy sequence, then there exists a vector la} E V such that
limi-->oo
lIa;
- a
II
=
o.
5.1.3. Example. 1.lll.iscompletewithrespectto theabsolute-value norm lIa
II
= la
I.
In
otherwords,everyCauchysequenceof real
numbers
hasalimitin
JR.
Thisisprovedin real
analysis.
2.
<C
is completewith respect to the norm
lIall
= lal =
~(Rea)2
+(rma)2. Using
lal
:s
I
Real
+ IImc], one can showthat the completeness of
<C
followsfrom that of R,
Detailsareleft as anexercisefor the
reader.
3. Theset of
rational
numbers
Qis not completewithrespectto theabsolute-value
norm.
In fact, {(I + 1/
k)k}~l
is a sequence
of
rational numbers that is Cauchy but does not
convergeto a
rational
number;
it converges to e, thebaseof thenaturallogarithm, whichis
known to bean irrational number. II
Let
{laj}}~l
be a Cauchy sequence
of
vectors in a finite-dimensional vec-
tor space
V
N.
Choose an orthonormal basis {lek)}f=l in VN such thatz laj} =
2Recallthatonecan
always
defineaninner
product
on a finite-dimensional vectorspace.So, theexistenceof orthonormal
basesis
guaranteed.
all
finite-dimensional
vector
spaces
are
complete
5.1
THE
QUESTION
OF
CONVERGENCE
147
N W N
ill
Lk=l
Cl
k
[ek)
and
[aj) =
Lk=t
Cl
k
[ek).
Then
lIa;
_ajll2
= (a;
-ajla;
-aj)
=
11~(Clf)
-Clk
j»)
lekf
N N
= L
(Clk;)
-
Clkj))*(Cl?)
- Cl?»)(ek[el) = L
[Clf)
-
Clk
j)
[2.
k,l=l k=l
The
LHS goes to zero, because the sequence is assumed Cauchy. Furthermore, all
terms on the RHS are positive. Thus, they too
must
go to zero as i, j -->
00.
By
the completeness
of
C, there
must
exist
Clk
E C such that
lim
n
....
oo
Clk
n)
=
Clk
for
k =
1,2,
...
, N. Now consider [a) E VN givenby la) =
Lf=l
Clk
lek). We claim
that [a) is the limit
of
the above sequence
of
vectors iu VN. Indeed,
We have proved the followiug:
5.1.4. Proposition. Every Cauchy sequence in afinite-dimensional innerproduct
space
overC
(or
R)
isconvergent. In otherwords, everyfinite-dimensionalcomplex
(or real) innerproduct space
is complete with respect to the norm induced by its
inner product.
Thenext
example
showshow
important
the
word
"finite"
is.
5.1.5.
Example.
Consider
{fk}~l'
theinfinitesequenceof continuousfunctions defined
iu theinterval
[-I,
+1] by
{
I
if
Ilk::ox::ol,
Ik(x)
= (kx + 1)/2
if
-llk::o
x::o
Ilk,
o
if
-I::ox::o
-Ilk.
This
sequence
belongsto
eO(_l,
1), the
inner
product
spaceof
continuous
functions
with
its usual inner product:
(II
g) =
J~l
!*(x)g(x)
dx.
It
is straightforwardto verify that
IIIk
-
/;11
2
=
J~ll!k(x)
- /;(x)1
2dx
.
,0.
Therefore,the sequence is
Cauchy.
k,j"""'*oo
However.
thelimit
of
this
sequence
is (see
Figure
5.1)
I(x)
=
{I
if
0< x <
I,
o if
-1<x<O,
whichis
discontinuous
atx = 0 and
therefore
does notbelongto the spacein whichthe
original sequence lies. III
148 5. HI
LBERT
SPACES
y
o
Figure5.1 The
limit
of the
sequence
of thecontinuous
functions
Ik
is a
discontinuous
function
that
is 1forx > 0 and0 forx <
O.
We see that infinite-dimensional vector spaces are not generally complete.
It
is
anontrivial taskto show
whether
ornot a given infinite-dimensional vectorspace
is complete.
Any vector space (finite- or infinite-dimensional) contains all finite linearcom-
binations of the form
I:7~1
a; lai)when it contains all the lai)'s.This follows from
the very definition of a vector space. However, the sitnation is different when
n
goes to infinity. For the vector space to contain the infinite sum, firstly, the mean-
ing
of
such a sum has to be clarified, i.e., a norm and an associated convergence
criterion needs to be put in place. Secondly, the vector space has to be complete
with respect tn that norm. A complete normed vector space is called a
Banach
Banach
space
space. We shall not deal with a general Banach space, but only with those spaces
whose norms arise natnrally from an inner product. This leads to the following
definition:
Hilbert
space
defined
5.1.6. Definition. A complete inner product space, commonly denoted by
11:,
is
called a Hilbert space.
Thus, all finite-dimensional real or complex vector spaces are Hilbert spaces.
However, when we speak
of
a Hilbert space, we shall usually assume that it is
infinite-dimensional.
It
is convenient to use orthonormal vectors in stndying Hilbert spaces. So, let
us consider an infinite sequence
(lei)
}~1
of
orthonormal vectors all belonging to
a Hilbert space
11:.
Next, take any vector
If)
E
11:,
construct the complex numbers
fi = (ei
If),
and form the sequence
of
vectors''
n
Ifn) = L fi lei)
;=1
for n =
1,2,
...
(5.1)
3WecanconsiderIfn)asan
"approximation"
to
If),
becauseboth
share
thesame
components
alongthesamesetof
orthonormal
vectors.
The
sequence
of
orthonormal
vectorsactsverymuchasabasis.
However,
to bea basis,anextra
condition
mustbemet.
Weshalldiscussthis
condition
shortly.
5.1
THE
QUESTION
OF
CONVERGENCE
149
For
the pair
of
vectors
If)
and
IJ.,},
the Schwarz inequality gives
(5.2)
whereEquation (5.1) has beenusedto evaluate
Unl
fn}. On the otherhand, taking
the inner product
of
(5.1) with
(fl
yields
n n n
Ulfn)
=
Lfdflei)
=
Lfd,'
= Ll.fil2.
i=1
;=1 ;=1
Parseval
inequality
Substitution
of
this in Equation (5.2) yields the
Parseval
inequality:
n
L
I.ti
1
2
:s
UI
f}
.
;=1
(5.3)
This conclusion is true for arbitrarily large n and can be stated as follows:
5.1.7.
Proposition.
Let Ilei)}?::t be an infinite set
of
orthonormal vectors in a
Hilbert space,
1[,
Let
If)
E
11:
and define complex numbers Ii = (eil
f).
Then
Bessel
inequality
the Besselinequality holds:
E?::l
Ifil
2
s UI
f).
The
Bessel inequality shows that the vector
00
n
" Ii lei)
==
lim
"Ii
lei)
L-
n--+ooL-
i=1
i=l
complete
orthonormal
sequence
of
vectors
converges; that is, it has a finitenorm. However, the inequalitydoes not say whether
the vector converges to
If).To make such a statement we need completeness:
5.1.8. Definition.
A sequence
of
orthonormal vectors I
lei)}?::!
in aHilbert space
11:
is called complete ifthe only vector in
11:
that is orthogonal to all the lei} is the
zero
vector.
This completeness property is the extra condition alluded to (in the footuote)
above, and is what is required to make a basis.
5.1.9.
Proposition.
Let
Ilei}}?::!
be an orthonormal sequence in
11:.
Then the
following statements are equivalent:
1.
Ilei}}?::!
is complete.
v
If)
E
1[,
2.
If)
=
E~!
[ei}
(eil
f)
3.
E~l
[ei}
(eil = 1.
4.
UI g) =
E?::l
UI
ei) (eil g)
v
If),
Ig}
E
11:.
150
5.
HILBERT
SPACES
v
If)
E 1f.
Proof
We shall prove the implications
1=}2
=} 3 =} 4 =} 5 =} I.
I =} 2:
It
is sufficient to show that the vector
11ft)
==
If)
- I:f;:! lei) (eil
f)
is
otthogonal to all the
Iej):
8ij
00
,..-'--,
(ejl1ft) =
(ejl
f)
- L
(ejl
ei) (eil
f)
=
O.
;=1
2 =} 3: Since
If)
=
llf)
=I:f;:j(lei) (eil)
If)
is true for all
If)
E
1C,
we must
have 1
=
I:f;:I!ei)(eil.
3 =} 4:
(fl
g)
=
(flllg)
=
(fl
(I:f;:I!ei)
(eil)
Ig)
=
I:f;:j
(fl
ei)(eil
g).
4 =} 5: Let
Ig}
=
If)
in statement 4 and recall that
(fl
ei) = (eil
f)*.
5 =} 1: Let
If)
be otthogonal to all the lei). Then all the terms in the sum are
zero implying that
1If11
2
= 0, which in
tum
gives
If)
= 0, because only the zero
vectorhas a zero
norm.
0
Parseval
equality;
generalized
Fourier
coefficients
The equality
00 00
IIff
=
(fl
f)
=L I(eil
f)
1
2
=L Ifil
2
,
;=1
i=1
fi = (eil
f)
,
(5.4)
is called the Parsevalequality,and the complexnumbers
Ji
are called generalized
completeness
Fourier coefficients. The relation
relation
00
1 = L lei} (eil
;=1
is called the completeness relation.
(5.5)
basis
for
Hilbert
5.1.10. Definition. A complete orthonormalsequence (lei)
Jf;:!
in aHilbertspace
spaces
1Cis called a basis of1C.
5.2 The Space of Square-IntegrableFunctions
Chapter 1 showed that the collection
of
all continuous fuoctions defined on an
interval
[a, b] forms a linear vector space. Example 5.1.5 showed that this space
is not complete. Can we enlarge this space to make it complete? Since we are
interested in an inner product as well, and since a natural inner product for func-
tions is defined in terms of integrals, we want to make sure that our fuoctions
are integrable. However, integrability does not require continuity, it only requires
piecewise continuity.
In
this section we shall discuss conditions under which the