Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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However, no matter how small  is, the

ratio P

regular

collapse

will approach 0 as V !1,

N=V ! v

1

; this occurs extremely rapidly because

bN

eventually dominates over V

 e

N log N

Thus, it is far more probable to find the system in a

microscopic volume of size  rather than in a

configuration in which the energy has some macro-

scopic value proportional to N. This catastrophe can

be called an ultraviolet catastrophe (as it is due to the

behavior at very short distances) and it causes the

collapse of the particles into configurations concen-

trated in regions as small as we please as V !1.

Coalescence Catastrophe due

to Long-Range Attraction

It occurs when the potential is too attractive near 1.

For simplicity, suppose that the potential has a hard

core, i.e., it is þ1 for r < r

, so that the above-

discussed coalescence cannot occur and the system

density bounded above by a certain quantity 

< 1

(close-packing density).

Thecatastropheoccursif’(q) gjqj

3þ"

, g, ">0,

for jqj large. For instance, this is the case for matter

interacting gravitationally; if k is the gravitational

constant, m is the particle mass, then g = km

and " = 2.

The probability P

regular

of ‘‘regular configurations,’’

where particles are at distances of order 

1=3

from

their close neighbors, is compared with the probability

collapse

of ‘‘catastrophic configurations,’’ with the

particles at distances r

from their close neighbors to

form a configuration of density 

=(1 þ )

almost in

close packing (so that r

is equal to the hard-core

radius times 1 þ ). In the latter case, the system does

not fill the available volume and leaves empty a region

whose volume is a fraction ((

 )=

)V of V.

Further, it can be checked that the ratio P

regular

collapse

tends to 0 at a rate O (exp (g

N(

(1 þ )

3

 )))

if  is small enough (and <

A system which is too attractive at infinity will not

occupy the available volume but will stay confined in a

close-packed configuration even in empty space.

This is important in the theory of stars: stars cannot

be expected to obey ‘ ‘r egular thermodynamics’’ and in

particular will not ‘ ‘evaporate’’ because their particles

interact via the gravitational force at large distances.

Stars do not occupy the whole volume given to them

(i.e., the universe); they do not collapse to a point only

because the interaction has a strongly repulsive core

(even when they are burnt out and the radiation pressure

is no longer able to keep them at a reasonable size).

Evaporation Catastrophe

This is another infrared catastrophe, that is, a

catastrophe due to the long-range structure of the

interactions in the above subsection; it occurs when

the potential is too repulsive at 1, that is,

’ðqÞþgjqj

3þ"

as q !1

so that the temperedness condition is again

violated.

In addition, in this case, the system does not

occupy the whole volume: it will generate a layer of

particles sticking, in close-packed configuration, to

the walls of the container. Therefore, if the density is

lower than the close-packing density, <

, the

system will leave a region around the center of the

container  empty; and the volume of the empty

region will still be of the order of the total volume of

the box (i.e., its diameter will be a fraction of the

box side L). The proof is completely analogous to

the one of the previou s case; except that now the

configuration with lowest energy will be the one

sticking to the wall and close packed there, rather

than the one close packed at the center.

Also this catastrophe is important as it is realized in

systems of charged particles bearing the same charge:

the charges adhere to the boundary in close-packing

configuration, and dispose themselves so that the

electrostatic potential energy is minimal. Therefore,

charges deposited on a metal will not occupy the whole

volume: they will rather form a surface layer minimiz-

ing the potential energy (i.e., so that the Coulomb

potential in the interior is constant). In general, charges

in excess of neutrality do not behave thermodynami-

cally: for instance, besides not occupying the whole

volume given to them, they will not contribute

normally to the specific heat.

Neutral systems of charges behave thermodyna-

mically if they have hard cores, so that the

ultraviolet catastrophe cannot occur or if they obey

quantum-mechanical laws and consist of fermionic

particles (plus possibly bosonic particles with

charges of only one sign).

For more details, we refer the reader to Lieb

and Lebowitz (1972) and Lieb and Thirring (2001).

Appendix 2: The Subadditivity Method

A simple consequence of the assumptions is that the

exponential in (5.2) can be bounded above by

BN

exp(



i = 1

) so that

1  Z

ð; ; VÞexp Ve



B

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2m

1



) 0 

log Z

ð; ; VÞe



B

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2m

1

½73

Consider, for simplicity, the case of a hard-core

interaction with finite range (cf. [14]). Consider a

Introductory Article: Equilibrium Statistical Mechanics 85

sequence of boxe s 

with sides 2

, where L

> 0

is arbitrarily fixed to be > 2R. The partition function

(, z) relative to the volume 

N¼0



dQe

FðQÞ

because the integral over the P variables can be

explicitly performed and included in z

if z is

defined as z = e



(2m

1

)

d=2

Then the box 

contains 2

boxes 

n1

for n  1

and

1  Z

 Z

n1

exp B2dðL

n1

=RÞ

d1



½74

because the corridor of width 2R around the

boundaries of the 2

cubes 

n1

filling 

has

volume 2RL

n1

and contains at most

n1

=R)

d1

particles, each of which interacts

with at most 2

other particles. Therefore,

p

def

log Z

 L

n1

log Z

n1

þ B

n

ðL

=RÞ

d1

for some 

> 0. Hence, 0  p

 p

n1

þ 

n

for some 

> 0andp

is bounded above and below

uniformly in n.So,thelimit[13] exists on the sequence

= L

and defines a function p

(, ).

A box of arbitrary size L can be filled with about

(L=L



)

boxes of side L



with



n so large that,

prefixed >0, jp

 p

j <for all n 



n. Likewise,

a box of size L

can be filled by about (L

=L)

boxes of size L if n is large. The latter remarks lead

us to conclude, by standard inequalities, that the

limit in [13] exists and coincides with p

The subadditivity method just demonstrated for

finite-range potentials with hard core can be extended

to the potentials satisfying just stability and tempered-

ness (cf. the section ‘‘Thermodynamic limit’’).

For more details, the reader is referred to Ruelle

(1969) and Gallavotti (1999).

Appendix 3: An Infrared Inequality

The infrared inequalities stem from Bogoliubov’s

inequality. Consider as an example the problem of

crystallization discussed in the section ‘‘Continuous

symmetries: ‘no d = 2 crystal’ theorem’’. Let hi

denote average over a cano nical equilibrium state

with Hamiltonian

H ¼

j¼1

þ UðQÞþ"WðQÞ

with given temperature and density parameters

, ,  = a

3

. Let {X, Y} =

X @

Y  @

X @

be the Poisson bracket. Integration by parts, with

periodic boundary conditions, yields



fC; Hgi  



fC; e

H

gdPdQ

Z

ð; ; NÞ



1

hfA



; Cgi ½75

as a general identity. The latter identity implies, for

A = {C, H}, that

hfH; Cg



fH; Cgi ¼ 

1

hfC; fH; C



ggi ½76

Hence, the Schwartz inequality hA



Aih{H, C}



{H, C}ijh{A



, C}ij

combined with the two

relations in [75], [76] yields Bogoliubov’s inequality:



Ai

1

jhfA



; Cgij

hfC; fC



; Hggi

½77

Let g, h be arbitrary complex (differentiable)

functions and @

= @

AðQÞ¼

def

j¼1

gðq

Þ; CðP; QÞ¼

def

j¼1

hðq

Þ½78

Then H =

þ F(q

, ..., q

), if

Fðq

; ...; q

Þ¼

j6¼j

’ðjq

 q

jÞ þ "

Wðq

so that, via algebra,

fC; Hg

ðh

F  p

ðp

 @

Þh

with h

def

h(q

). If h is real valued, h{C,{C



, H}}i

becomes, again via algebra,

 @

FðQÞ

þ "

Wðq

Þþ



ð@

(integrals on p

just replace p

by 2

1

and

h(p

)

i= 

1



i, i

). Therefore, the average

h{C,{C



, H}}i becomes

ðh

 h

’ðjq

 q

jÞ

þ"

Wðq

Þþ4

1

ð@

½79

Choose g(q)  e

i(kþK)q

, h(q) = cos q  k and

bound (h

 h

)

by k

 q

)

,(@

)

by k

and

86 Introductory Article: Equilibrium Statistical Mechanics

by 1. Hence [79] is bounded above by ND(k)

with

DðkÞ¼

def

4

1

j6¼j

ðq

q

j’ðq

q

Þj

þ"

jWðq

Þj

½80

This can be used to estimate the denominator in

[77]. For the LHS remark that



; Ai¼j

j¼1

iqðkþKÞ

and

jhfA



; Cgij



¼jK þ kj

ð

ðKÞþ

ðK þ 2kÞÞ

hence [77] becomes, after multiplying both sides

by the auxiliary function (k) (assumed even and

vanishing for jkj >=a) and summing over k,

def

ðkÞ

j¼1

iðKþk Þq



ðkÞ



jKj

4

ð

ðKÞþ

ðK þ 2kÞÞ

DðkÞ

½81

To apply [77] the averages in [80], [81] have to be

bounded above: this is a technical point that is

discussed here, as it illustrates a general method of

using the results on the thermodynamic limits and

their convexity properties to obtain estimates.

Note that h(1=N)

(k)d

j = 1

ikq

i is

identically

’(0) þ (2=N)h

j<j

’(q

 q

)i with

’(q) =

def

(1=N)

(k)e

ikq

Let ’

, 

(q) =

def

’(q) þ q

j’(q)jþ

’(q) and

let F

(, , ) =

def

(1=N) log Z

(, , ) with Z

the

partition function in the volume  computed

with energy U

’

, 

 q

) þ "

W(q

) þ

"

jW(q

)j. Then F

(, , ) is convex in , 

and it is uniformly bounded above and below if

jj, j"j, jj1 (say) and jj

: here 

> 0 exists

if r

j’(r)j satisfies the assumption set at the

beginning of the section ‘‘Continuous symmetries:

‘no d = 2 crystal’ theorem’’ and the density is smaller

than a close packing (this is because the potential U

will still satisfy conditions similar to [14] uniformly

in j"j, jj < 1 and jj small enough).

Convexity and boundedness above and below

in an interval imply bounds on the derivatives in

the interior points, in this case on the derivatives of F

with respect to , ,  at 0. The latter are identic al to

the averages in [80], [81]. In this way, the constants

, B

such that D(k)  k

þ "B

and B

> D

are found.

For more details, the reader is refe rred to Mermin

(1968).

Further Reading

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fields and Ising

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Aizenman M, Lieb EH, Seiringer R, Solovej JP, and Yngvason J

(2004) Bose–Einstein condensation as a quantum phase

transition in a optical lattice, Physical Review A 70: 023612.

Baxter R (1982) Exactly Solved Models. London: Academic Press.

Benfatto G and Gallavotti G (1995) Renormalization group.

Princeton: Princeton University Press.

Bleher P and Sinai Y (1975) Critical indices for Dyson’s asympto-

tically hierarchical models. Communications in Mathematical

Physics 45: 247–278.

Boltzmann L (1968a) U

ber die mechanische Bedeutung des zweiten

Haupsatzes der Wa¨ rmetheorie. In: Haseno¨hrl F (ed.) Wissensch af-

tliche Abhandlungen,vol.I,pp.9–33.NewYork:Chelsea.

Boltzmann L (1968b) U

ber die Eigenshaften monzyklischer und

anderer damit verwandter Systeme. In: Haseno¨ hrl FP (ed.)

Wissenshafltliche Abhandlungen, vol. III, pp. 122–152.

New York: Chelsea.

Dobrushin RL (1968) Gibbsian random fields for lattice systems

with pairwise interactions. Functional Analysis and Applica-

tions 2: 31–43.

Domb C and Green MS (1972) Phase Transitions and Critical

Points. New York: Wiley.

Dyson F (1969) Existence of a phase transition in a one–dimensional

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Berlin: Springer.

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two dimensions. Physical Review 158: 383–386.

Introductory Article: Equilibrium Statistical Mechanics 87

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MIR.

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states with short range correlations in statistical mechanics.

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in Mathematical Physics 28: 313–321.

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Introductory Article: Functional Analysis

S Paycha, Universite

Blaise Pascal, Aubie

re, France

Introduction

Functional analysis is concerned with the study of

functions and function spaces, combining techniques

borrowed from classical analysis with algebraic

techniques. Modern functional analysis developed

around the problem of solving equations with

solutions given by functions. After the differential

and partial differential equations, which were

studied in the eighteenth century, came the integral

equations and other types of functional equations

investigated in the nineteenth century, at the end of

which arose the need to develop a new analysis,

with functions of an infinite number of variables

instead of the usual functions. In 1887, Volterra,

inspired by the calculus of variations, suggested a

new infinitesimal calculus where usual functions are

replaced by functionals, that is, by maps from a

function space to R or C, but he and his followers

were still missing some algebraic and topological

tools to be developed later. Modern analysis was

born with the development of an ‘‘algebra of the

infinite’’ closely related to classical linear algebra

which by 1890 had (up to the concept of duality,

which was developed later) settled on firm ground.

Strongly inspired by algebraic methods, Fredholm’s

work at the turn of the nineteenth century, in which

emerged the concept of kernel of an operator,

became a founding stone for the modern theory of

integral equations. Hilbert developed further Fred-

holm’s methods for symmetric kernels, exploiting

analogies with the theory of real quadratic forms

and thereby making clear the importance of the

notion of square-integrable functions. With Hilbert’s

Grundzu¨ ge einer allgemeinen Theorie der Integral-

gleichung, a further step was made from the

‘‘algebra of the infinite’’ to the ‘‘geometry of the

infinite.’’ The contribution of Fre´chet, who intro-

duced the abstract notion of a space endowed with a

distance, made it possible to transfer Euclidean

geometry to the framework of what have since

then been called Hilbert spaces, a basic concept in

mathematics and quantum physics.

The usefulness of functional analysis in the study

of quantum systems became clear in the 1950s when

Kato proved the self-adjointness of atomic Hamilto-

nians, and Garding and Wightman formulated

axioms for quantum field theory. Ever since func-

tional analysis lies at the very heart of many

approaches to quantum field theory. Applications

of functional analysis stretch out to many branches

of mathematics, among which are numerical

88 Introductory Article: Functional Analysis

analysis, global analysis, the theory of pseudodiffer-

ential operators, differential geometry, operator

algebras, noncommutative geometry, etc.

Topological Vector Spaces

Most topological spaces one comes across in practice

are metric spaces. A metric on a topological space E

is a map d : E  E ! [0, þ1[ which is symmetric,

such that d(u, v) = 0 , u = v and which verifies the

triangle inequality d(u, w)  d(u, v) þ d(v, w) for all

vectors u, v, w.AtopologicalspaceE is metrizable if

there is a metric d on E compatible with the topology

on E, in which case the balls with radius 1=n centered

at any point x 2 E form a local base at x – that is, a

collection of neighborhoods of x such that every

neighborhood of x contains a member of this

collection. A sequence (u

)inE then converges to

u 2 E if and only if d(u

, u) converges to 0.

The Banach fixed-point theorem on a complete

metric space (E, d) is a useful tool in nonlinear

functional analysis: it states that a (strict) contrac-

tion on E, that is, a map T : E ! E such that

d(Tu , Tv)  k(u, v) for all u 6¼ v 2 E and fixed 0 <

k < 1, has a unique fixed point T u

= u

.In

particular, it provides local existence and uniqueness

of solutions of differential equations dy= dt = F(y, t)

with initial condition y(0) = y

, where F is Lipschitz

continuous.

Linear functional analysis starts from topological

vector spaces, that is, vector spaces equipped with a

topology for which the operations are continuous. A

topological vector space equipped with a local base

whose members are convex is said to be locally

convex. Examples of locally convex spaces are

normed linear spaces, namely vector spaces

equipped with a norm, a concept that first arose in

the work of Fre´chet. A seminorm on a vector space

V is a map  : V ! [0,1[ which obeys the triangle

identity (u þ v)  (u) þ (v) for any vectors u, v

and such that (u) = jj(u) for any scalar  and

any vector u;if(u) = 0 ) u = 0, it is a norm, often

denoted by kk. A norm on a vector space E gives

rise to a translation-invariant distance function

d(u , v) = ku  vk making it a metric space.

Historically, one of the first examples of normed

spaces is the space C([0, 1]) investigated by Riesz of

(real- or complex-valued) continuous functions on

the interval [0, 1] equipped with the supremium

norm kf k

:= sup

x2[0,1]

jf (x)j. In the 1920s, the

general definition of Banach space arose in connec-

tion with the works of Hahn and Banach. A normed

linear space is a Banach space if it is complete as a

metric space for the induced metric, C([0, 1]) being a

prototype of a Banach space. More generally, for

any non-negative integer k, the space C

([0, 1]) of

functions on [0, 1] of class C

equipped with the

norm kf k

i = 0

(i)

expressed in terms of a

finite number of seminorms kf

(i)

= sup

x2[0,1]

(i)

(x)j, i = 0, ..., k, is also a Banach space.

The space C

([0, 1]) of smooth functions on the

interval [0, 1] is not anymore a Banach space since

its topology is described by a countable family of

seminorms kf k

with k varying in the positive

integers. The metric

dðf ; gÞ¼

k¼1

k

kf  gk

1 þkf  gk

turns it into a Fre´chet space, that is, a locally convex

complete metric space. The space S(R

) of rapidly

decreasing functions, which are smooth functions f

on R

for which

kf k

;

:¼ sup

x2R





f ðxÞj

is finite for any multiindices  and , is also a

Fre´chet space with the topology given by the

seminorms kk

, 

. Further examples of Fre´chet

spaces are the space C

(K) of smooth functions

with support in a fixed compact subset K  R

equipped with the countable family of seminorms



f k

1; K

¼ sup

x2K



f ðxÞj;2 N

and the space C

(M, E) of smooth sections of a

vector bundle E over a closed manifold M equipped

with a similar countable family of seminorms. Given

an open subset = [

p2N

with K

, p 2 N com-

pact subsets of R

, the space D() = [

p2N

)

equipped with the inductive limit topology – for

which a sequence (f

)inD() converges to f 2D()

if each f

has support in some fixed compact subset

K and (D



) converges uniformly to D



f on K for

each mutilindex  – is a locally convex space.

Among Banach spaces are Hilbert spaces which

have properties very similar to those of finite-

dimensional spaces and are historically the first

type of infinite-dimensional space to appear with the

works of Hilbert at the beginning of the twentieth

century. A Hilbert space is a Banach space equippe d

with a norm kk that derives from an inner product,

that is, kuk

= hu, ui with h, i a positive-definite

bilinear (or sesquilinear according to whether the

base space is real or complex) form. Hilber t spaces

are fundamental building blocks in quantum

mechanics; using (closed) tensor products, from a

Hilbert space H one builds the Fock space

F(H) =

k = 0



H and from there the bosonic

Fock space F(H) =

k = 0



H (where 

stands

for the (closed) symmetrized tensor product) as well

Introductory Article: Functional Analysis 89

as the fermionic Fock space F(H) =

k = 0



(where 

stands for the antisymmetrized (closed)

tensor product).

A prototype of Hilbert space is the space l

(Z)of

complex-valued sequences (u

)

n2Z

such that

n2Z

is finite, which is already implicit in

Hilbert’s Grundzu¨ gen. Shortly afterwords, Riesz and

Fischer, with the help of the integration tool

introduced by Lebesgue, showed that the space

(]0, 1[) (first introduced by Riesz) of square-

summable functions on the interval ]0, 1[, that is,

functions f such that

kf k

jf ðxÞj



1=2

is finite, provides an example of Hilbert space.

These were then further generalized to spaces

(]0, 1[) of p-summable (1  p < 1) funct ionals

on ]0, 1[ (i.e., functions f such that

kf k

jf ðxÞj



1=p

is finite), which are not Hilbert unless p = 2 but which

provide further examples of Banach spaces, the space

(]0, 1[) of functions on ]0, 1[ bounded almost

everywhere with respect to the Lebesgue measure,

offering yet another example of Banach space.

In 1936, Sobolev gave a generalization of the

notion of function and their derivatives through

integration by parts, which led to the so-called

Sobolev spaces W

k, p

(]0, 1[) of functions f 2

(]0, 1[) with derivatives up to order k lying in

(]0, 1[), obtained as the closure of C

(]0, 1[) for

the norm

f 7!kf k

k;p

j¼1

f k

1=p

(for p = 2, W

k, p

(]0, 1[) is a Hilbert space often

denoted by H

(]0, 1[). They differ from the Sobolev

spaces W

k, p

(]0, 1[), which correspond to the closure

of the set D(]0, 1[) for the norm f 7!kf k

k, p

; for

example, an element u 2 W

1, p

(]0, 1[) lies in

1, p

(]0, 1[) if and only if it vanishes at 0 and 1,

that is, if and only if it satisfies Dirichlet-type

boundary conditions on the boundary of the inter-

val. Similarly, one defines Sobolev spaces

k, p

(R) = W

k, p

(R)onR, Sobolev spaces W

k, p

()

and W

k, p

() on open subsets   R

and using a

partition of unity on a closed manifold M, Sobolev

spaces H

(M, E) = W

k,2

(M, E) of sections of vector

bundles E over M. Using the Fourier transform

(discussed later), one can drop the assumption that k

be an integer and extend the notion of Sobolev space

to define W

s, p

() and H

(M, E) with s any real

number.

Sobolev spaces arise in many areas of mathe-

matics; one central example in probability theory is

the Cameron–Martin space H

([0, t]) embedded in

the Wiener space C([0, t]). This embedding is a

particular case of more general Sobolev embedding

theorems, which embe d (possibly continuously,

sometimes even compactly (the notion of compact

operator is discussed in a later section)) W

k, p

Sobolev spaces in L

-spaces with q > p such as the

continuous inclusion W

k, p

)  L

) with

1=q = 1=p  k=n,orinC

-spaces with l  k such

as, for a bounded open and regular enough subset 

of R

and for any s  l þ n=p with p > n, the

continuous inclusion W

s, p

()  C

(



) (the set of

functions in C

() such that D



u can be continu-

ously extended to the closure



 for all jjl).

Sobolev embeddings have important applications for

the regularity of solutions of partial differential

equations, when showing that weak solutions one

constructs are in fact smooth. In parti cular, on an n-

dimensional closed manifold M for s > l þ n=2, the

Sobolev space H

(M, E) can be continuously

embedded in the space C

(M, E) of sections of E of

class C

, which in particular implies that the

solutions of a hypoelliptic partial differential equa-

tion Au = v with v 2 L

(M, E) are smooth, as for

example in the case of solutions of the Seiberg–

Witten equations.

Duality

The concept of duality (in a topological sense) was

initiated at the beginning of the twentieth century by

Hadamard, who was looking for continuous linear

functionals on the Banach space C(I) of continuous

functions on a compact interval I equipped with a

uniform topology. It is implicit in Hilbert’s theory

and plays a central part in Riesz’ work, who

managed to express such continuous functionals as

Stieltjes integrals, one of the starting points for the

modern theory of integration.

The topological dual of a topological vector space

E is the space E



of continuous linear forms on E

which, when E is a normed space, can be equipped

with the dual norm kLk



= sup

u2E, kuk1

jL(u)j.

Dual spaces often provide a receptacle for singular

objects; any of the functions f 2 L

)(p  1) and

the delta-function at point x 2 R

, 

: f 7!f (x), all lie

in the space S

) dual to S(R

) of tempered

distributions on R

, which is itself contained in the

space D

) of distributions dual to D(R

Furthermore, the topological dual E



of a nuclear

space E contains the support of a probability

90 Introductory Article: Functional Analysis

measure with characteristic function (see the next

section) given by a continuous positive-definite

function on E. Among nuclear spaces are projective

limits E = \

p2N

(a sequence (u

) 2 E converges

to u 2 E whenever it converges to u in each H

)of

countably many nested Hilbert spaces H



p1

H

such that the embedding H



p1

is a trace-class operator (see the section

‘‘Op erator alge bras’’). If H

is the closure of E for

the norm kk

, the topological dual E

of E for the

norm kk

is an inductive limit E

= [

p2N

p

where H

p

are the dual (with respect to kk

)

Hilbert spaces with norm kk

p

(a sequence (u

) 2

converges to u 2 E

whenever it lies in some H

p

and converges to u for the topology of H

p

) and we

have

E H

 H

p1

H

¼ H

 H

1

H

p

E

As a result of the theory of elliptic operators on a

closed manifold, the Fre´chet space C

(M, E)of

smooth sections of a vector bundle over a closed

manifold M is nuclear as the inductive limit of

countably many Sobolev spaces H

(M, E) with

-dual given by the projective limit of countably

many Sobolev spaces H

p

(M, E).

The existence of nontrivial continuous linear

forms on a normed linear space E is ensured by the

Hahn–Banach theorem, which asserts that for any

closed linear subspace F of E, there is a nonvanish-

ing continuous linear form that vanishes on F. When

the space is a Hilbert space (H,h, i

), it follows

from the Riesz–Fre´chet theorem that any continuous

linear form L on H is represented in a unique way

by a vector v 2 H such that L(u) = hv, ui

for all

u 2 H, thus relating the dua l pairing on the left with

the Hilbert inner product on the right and identify-

ing the topological dual H



with H.

The strong topology induced by the norm kkon

a normed vector space E – that is, the topology in

which a sequen ce (u

) converges to u whenever

 uk!0 – is too refined to have compact sets

when E is infinite dimensional since the compactness

of the unit ball in E for the strong topology

characterizes finite-dimensional spaces. Since com-

pact sets are useful for existence theorems, one is

inclined to weaken the topology: the weak topology

on E – which coincides with the strong topology

when E is finite dimensional and for which a

sequence (u

)convergestou if and only if L(u

) !

L(u) 8L 2 E



– has compact unit ball if and only if E

is reflexive or, in other words, if E can be canonically

identified with its double dual (E



)



.For1< p < 1,

given an open subset   R

, the topological dual of

() can be identified via the Riesz representation

with L



()withp



conjugate to p,thatis,1=p þ

1=p



= 1andL

() is reflexive, whereas the topolo-

gical duals of W

s, p

()andW

s, p

() both coincide

with W

s, p



() so that only W

s, p

()isreflexive.

Neither L

() nor its topological dual L

()is

reflexive since L

() is strictly contained in the

topological dual of L

() for there are continuous

linear forms L on L

() that are not of the form

LðuÞ¼



uv 8u 2 L

ðÞ with v 2 L

ðÞ

Similarly, the topological dual E



of a normed

linear space E can be equipped with the topology

induced by the dual norm kk



and the the weak -

topology, namely the weakest one for which the

maps L 7!L(u), u 2 E, are continuous, and the unit

ball in E



is indeed compact for this topology

(Banach–Alaoglu theore m).

Duality does not always preserve separability – a

topological vector space is separable if it has a

countable dense subspace – since L

(), which is

not separable, is the topological dual of L

(),

which is separable. However, as a consequence of

the Hahn–Banach theorem, if the topological dual of

a Banach space is separable then so is the original

space and one has equivalence when adding the

reflexivity assumption; a Banach space is reflexive

and separable whenever its topological dual is. For

1  p < 1, L

() and W

s, p

() are separable and

moreover reflexive if p 6¼ 1.

Fourier Transform

In the middle of the eighteenth century, oscillations

of a vibrating string were interpreted by Bernouilli

as a limit case for the oscillation of n-point masses

when n tends the infinity, and Bernouilli introduced

the novel idea of the superposition principle by

which the general oscillation of the string should

decompose in a superpos ition of ‘‘proper oscilla-

tions.’’ This point of view triggered off a discussion

as to whether or not an arbitrary function can be

expanded as a trigonometric series. Other examples

of expansions in ‘‘orthogonal functions’’ (this termi-

nology actually only appears with Hilbert) had been

found in the mean time in relation to oscillation

problems and investigations on heat theory, but it

was only in the nineteenth century, with the works

of Fourier and Dirichlet, that the superposition

problem was solved.

Separable Hilbert spaces can be equipped with a

countable orthonormal system {e

}

n2Z

(he

, e



with h, i

the scalar product on H) which is

Introductory Article: Functional Analysis 91

complete, that is, any vector u 2 H can be expanded

in this system in a unique way u =

n2Z

with

Fourier coefficie nts

= hu, e

i. The latter obey

Parseval’s relation

n2Z

= kuk

(where kk is

the norm associated with h, i), and the Fourier

transform u 7!(

u(n))

n2Z

gives rise to an isometric

isomorphism between the separable Hilbert space

H and the Hilbert space l

(Z) of square-summable

sequences of complex numbers. In particular, the

space L

)ofL

-functions on the unit circle

= R=Z with its usual Haar measure dt is separ-

able with complete orthonormal system t 7!e

(t) =

2int

, n 2 Z and the Fourier transform

u 7! t 7!

uðnÞ¼

2int

uðtÞdt



n2Z

identifies it with the space l

(Z). Under this

identification, the Hilbert subspace l

(N) obtained

as the range in l

(Z) of the projection p

:(u)

n2Z

)

n2N

corresponds to the Hardy space H

The Fourier transform extends to the space S(R

sending a function f 2S(R

) to the map

7!

f ðÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð2Þ

ix

f ðxÞdx

and maps S(R

) onto itself linearly and continuously

with continuous inverse f 7!

f (). Wh en n = 1, the

Poisson formula relates f 2S(R) with its Fourier

transform

f by

n = 1

f (2n) =

n = 1

f (n).

Since Fourier transformation turns (up to a

constant multiplicative factor) differentiation D





for a multiindex  = (

, ..., 

) into multiplication

by 



= 







, it can be used to define W

s, p

Sobolev spaces with s a real number as the space of

-functions with finite Sobolev norms kuk

s, p

(

j(1 þjj)

u( )j

)

1=p

(which coincide with the ones

defined previously when s = k is a non-negative

integer).

Fourier t ransforms are also used to describe a

linear pseudodifferential operator A (see ne xt two

sections where the notions of bounded and

unbounded linear operator are discussed) of order

a acting on smooth functions on an open subset U

of R

in terms of its symbol 

– a smooth map 

on U  R

with compact support in x such that for

any multi-indic es ,  2 N

, there is a constant

,

with







ðx;ÞC

;

ð1 þjjÞ

ajj

for any  2 R

–by

ðAf Þðx Þ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð2Þ

ix



ðx;Þ

f ðÞd

Fourier transform maps a Gaussian function

x 7!e

(1=2)jxj

on R

, where  is a nonzero scalar,

to another Gaussian function  7!e

(1=2)

1

jj

(up to

a nonzero multiplicative factor), a starting point for

T-duality in string theory. More generally, the

characteristic function

ðÞ :¼

ihx;i

ðdxÞ

of a Gaussian probability measure  with covariance

C on a Hilbert space H is the funct ion

 7!e

(1=2)h, Ci

. Such probability measures typically

arise in Euclidean quantum field theory; in axio-

matic quantum field theory, the analyticity proper-

ties of n-point functions can be derived from the

Wightman axioms using Fourier transforms. Thus,

Fourier transformation underlies many different

aspects of quantum field theory.

Fredholm operators

A complex-valued continuous function K on [0, 1] 

[0, 1] give s rise to an integral operator

A : f !

Kðx; yÞf ðyÞdy

on complex-valued continuous functions on [0, 1]

(equipped with the supremum norm kk

) with the

following upper bound property:

kAfk

 Sup

½0;1½0;1

jKðx; yÞjkf k

In other words, A is a bounded linear operator with

norm bounded from above by sup

[0, 1][0, 1]

jK(x, y)j;

a linear operator A : E ! F from a normed linear

space (E,kk

) to a normed linear space (F,kk

)is

bounded (or conti nuous) if and only if its (operator)

norm jkAkj:= sup

kuk

1

kAuk

is bounded.

An integral operator

A : f !

Kðx; yÞf ðyÞdy

defined by a continuous kernel K is, moreover,

compact; a compact operator is a bounded operator

of normed spaces that maps bounded sets to a

precompact sets, that is, to sets whose closure is

compact. Other examples of compact operators on

normed spaces are finite-rank operat ors, operators

with finite-dimensional range. In fact, any compact

operator on a separable Hilbert space can be

approximated in the topology induced by the

operator norm jk kj by a sequence of finite-rank

operators.

Inspired by the work of Volterra, who, in the case

of the integral operat or defined above, pro duced

92 Introductory Article: Functional Analysis

continuous solutions  = (I  A)

1

f of the equation

f = (I  A) for f 2 C([0, 1]), Fredholm in 1900

(Su r une classe d’e´quations fonctionnelles) studied the

equation f = ( I   A) , introducing a complex para-

meter . He proved what is since then called the

Fredholm alternative, which states that either the

equation f = (I  A) has a unique solution for every

f 2 C([0, 1]) or the corresponding homogeneous equa-

tion (I  A) = 0 has nontrivial solutions. In modern

language, it means that the resolvent R(A, ) = (A 

I)

1

of a compact linear operator A is surjective if and

only if it is injective. The Fredholm alternative is a

powerful tool to solve partial differential equations

among which the Dirichlet problem, the solutions of

which are harmonic functions u (i.e., u = 0, where

=

i = 1

u=@x

) on some domain  2 R

with

Dirichlet boundary conditions u

@

= f ,wheref is a

continuous function on the boundary @.TheDirichlet

problem has geometric applications, in particular to the

nonlinear Plateau problem, which minimizes the area of

asurfaceinR

with given boundary curves and which

reduces to a (linear) Dirichlet problem.

The operator B = I  A built from the compact

operator A is a particular Fredholm operator, namely a

bounded linear operator B : E ! F which is invertible

‘‘up to compact operators,’’ that is, such that there is a

bounded linear operator C : F ! E with both BC  I

and CB  I

compact. A Fredholm operator B has a

finite-dimensional kernel Ker B and when (E,h, i

)

and (F,h, i

) are Hilbert spaces its cokernel Ker B



where B



is the adjoint of B defined by

hBu; vi

¼hu; B



8u 2 E; 8v 2 F

is also finite dimensional, so that it has a well-

defined index ind(B) = dim(Ker B)  dim(Ker B



), a

starting point for index theory. To¨ plitz operators



, where  is a continuous function on the unit

circle S

, provide first examples of Fredholm

operators; they act on the Hardy space H

)by

n

m0

mþn

under the identification H

) ’ l

(N)  l

(Z),

with l

(Z) equipped with the canonical complete

orthonormal basis (e

, n 2 Z). The Fredholm index

ind(T

n

) is exactly the integer n so that the index of

its adjoint is n, as a consequence of which the index

map from Fredholm operat ors to integers is onto.

One-Parameter (Semi) groups

Unlike in the finite-dimensional situation, a linear

operator A : E ! F between two normed linear

spaces (E,kk

) and (F,kk

) is not expected to be

bounded. Unbounded operators arise in partial

differential equations that involve differential opera-

tors such as the Laplacian  on an open subset  

. The following equations provide fundam ental

examples of partial differential equations which

arose over time from the study of various problems

in mathematical physics with the works of Poisson,

Fourier, and Cauchy:

u ¼ 0 Laplace equation

þ u ¼ 0 wave equation

þ u ¼ 0 heat equation

and later the Schro¨ dinger equation in quantum

mechanics:

¼ u

where t is a time parameter.

An unbounded linear operator on an infinite-

dimensional normed space is usually defined on a

domain D(A) which is strictly contained in E. The

Laplacian  is defined on the dense domain

D(A) = H

)inL

); it defin es a bounded

operator from H

)toL

) but does not

extend to a bounded operator on L

). Like this

operator, most unbounded operators A : E ! F one

comes across have dense domain D(A)inE and are

closed, that is, their graph {(u, Au), u 2 D(A )} is

closed as a subset of the normed linear space E  F.

When not actually closed, they can be closable, that

is, they can have a closed extension called the

closure of the operator. By the closed-graph theo-

rem, when E and F are Banach spaces, a linear

operator A : E ! F is continuous whenever its graph

is closed, as a consequence of which a closed linear

operator A : E ! F defined on a dense domain is

bounded provided its domain coincides with the

whole space.

For a closed operator A : E ! F with dense

domain D(A), when E and F are Hilbert spaces

equipped with inner products h, i

and h, i

, the

adjoint A



of A is defined on its domain D(A



)by

hAu; vi

¼hu; A



8ðu; vÞ2DðAÞDðA



A self-adjoint operator A with domain D(A) is one

for which D(A) = D(A



) and A = A



; the Laplacian

 on R

is self-adjoint on the Sobolev space H

)

but it is only essentially self-adjoint on the dense

domain D(R

), the latter meani ng that its closure is

self-adjoint.

Unbounded self-ad joint operators can arise as

generators of one-parameter semigroups of bounded

Introductory Article: Functional Analysis 93

operators. A one-parameter family of bounded

operators T

, t  0(T

, t 2 R) on a Hilbert space H

is a semigroup (resp. group) if T

= T

tþs

8t , s  0

(resp. 8t, s 2 R) and it is strongly continuous (or

simply continuous) if lim

t !t

u = T

u at any t

 0

(resp. t

2 R) and for any u 2 H.

Stones’ theorem sets up a one-to-one correspon-

dence between continuous one-parameter unitary



= U



= I) groups U

, t 2 R on a Hilbert

space such that U

= Id and self-adjoint operators

A obtained as infinitesimal generators, that is, as the

strong limit

Au ¼ lim

t!0

u  u

; u 2 H

of U

, t 2 R, which in a compact form reads

= e

itA

. An important example in quantum

mechanics is U

= e

itH

, t 2 R with H a self-

adjoint Hamiltonian, which solves the Schro¨ dinger

equation d=dtu = iHu. The Lie–Trotter formula,

which has important app lications for Feynman

path integrals, expresses the unita ry semigroup

generated by A þ B, where A, B, and A þ B are

self-adjoint on their respective domains as a strong

limit

itðAþBÞ

¼ lim

t!1

itA

itB



On the other hand, positive operators on a

Hilbert space ( H,h, i

) – that is, A self-adjoint

and such that hAu, ui

 0 8u 2 D(A) – generate

one-parameter semigroups T

= e

tA

, t  0. Hille

and Yosida proved that on a Hilbert space, strongly

continuous contraction (i.e., jkT

kj  1 8t > 0)

semigroups such that T

= Id are in one-to-one

correspondence with densely defined positive opera-

tors A : D(A)  H ! H that are maximal (i.e., I þ A

is onto), obtained as (minus the) infinitesimal

generators

Au ¼ lim

t!0

u  u

; u 2 H

of the corresponding semigroups. Similarly, a posi-

tive densely defined self-adjoint operator A on a

Hilbert space H gives rise to a densely defined closed

symmetric sesquilinear form (u, v) 7!h

ﬃﬃﬃﬃ

(see next section for a definition of

ﬃﬃﬃﬃ

;h, i

is the

scalar product on H) and this map yields a one-

to-one correspondence between operators and

sesquilinear forms on H with the aforementioned

properties, one of the starting points for the theory

of Dirichlet forms. To a probability measure  on

a separable Banach space E, one can associate a

densely defined closed symmetric sesquilinear form

(it is in fact a Dirichlet form) on a Hilbert space H

such that E



 H



= H  E, which in the particular

case of the standard Wiener measure  on the

Wiener space E = C ([0, t]) and with Hilbert space

given by the Cameron–Martin space H = H

([0, t]),

is the bilinear form

ðu; vÞ7!



ru;



rvi

with



r the (closed) gradient of Malli avin calculus.

The operator ,where is the Laplacian on R

generates the heat-operator semigroup e

t

, t  0. It

has a smooth kernel K

2 C

 R

) defined by

ðe

t

f ÞðxÞ¼

ðx; yÞf ðyÞdy 8f 2 C

ðR

and defines a smoothing operator, an operator that

maps Sobolev function to smooth function. In

general, a pseudodifferential operators A on an

open subset U of R

with symbol 

only has a

distribution kernel

ðx; yÞ¼

ihxy;i

ð Þd

The kernel of the inverse Laplacian ( þ m

)

1

on R

(the non-negativ e r eal n umber m

stands

for the mass) called Green’s function on R

plays an essential role in the theory o f Feynman

graphs.

Spectral Theory

Spectral theory is the study of the distribution of the

values of the complex parameter  for which, given

a linear operator A on a normed space E, the

operator A  I has an inverse and of the properties

of this inverse when it exists, the resolvent

R(A, ) = (A  I)

1

of A. The resolvent (A)ofA

is the set of complex numbers  for which A  I is

invertible with densely defined bounde d inverse. The

spectrum Sp(A)ofA is the complement in C of the

resolvent; it consists of a union of three disjoint sets:

the set of all complex numbers  for which A  I is

not injective, called the point spectrum – such a  is

an eigenvalue of A with associated eigenfunction

any u 2 D(A) such that Au = u; the set of points 

for which A  I has a densely defined unbounded

inverse R(A, ) called the continuous spectrum; and

the set of points  for which A  I has a well-

defined unbound ed but not densely defined inverse

R(A, ) called the residual spectrum.

A bounded operator has bounded spectrum and a

self-adjoint operator A acting on a Hilbert space has

real spectrum and no residual spectrum since the

range of A  I is dense. As a consequence of the

94 Introductory Article: Functional Analysis