Spaces of analytic functions on inductive/projective limits of Hilbert spaces

Spaces of analytic functions on inductive/projective limits of

Hilbert spaces

Citation for published version (APA):

Martens, F. J. L. (1988). Spaces of analytic functions on inductive/projective limits of Hilbert spaces. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR292007

DOI:

10.6100/IR292007

Document status and date: Published: 01/01/1988

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ON

INDUCTIVE/PROJECTIVE LIMITS

OF

HILBERT SPACES

ON

INDUCTIVE/PROJECTIVE LIMITS

OF

HILBERT SPACES

PROEFSCHRIFf

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE TECHNISCHE UNIVERSITEIT EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. IR. M. TELS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN

OP DINSDAG 8 NOVEMBER 1988 TE 14.00 UUR DOOR

FRANCISCUS JOHANNES LUDOVICUS MARTENS

GEBOREN TE LITH

Prof. dr.ir. J. de Graaf en

Prof. dr. S.Th.M. Ackermans Copromotor:

General introduction

I Preliminaries

§ 1. Locally convex spaces

§ 2. Inductive and projective limits § 3. Sequences and sequence sets § 4. Analytic functions

II Functional Hilbert spaces Introduction i 6 9 14 27 27

§ 1. Reproducing kemel theory 29

§ 2. Synnnetric Fock spaces as functional Hilbert spaces 44 § 3. Examples of functional Hilbert spaces 55

Appendix 71

III Inductive and projective limits of semi-inner product spaces 77 Introduction

§ 1. Positive sequence sets

§ 2. Hilbertian dual systeins of inductive limits and projective limits

§ 3. Cross synnnetric moulding sets

Appendix

IV Spaces of analytic functions on sequence spaces Introduction § 1. Generating sets 77 79 85 100 103 109 109 112 § 2. Sequence spaces 120

§ 3. A sequence space representation of A(qi(]))) and A(w(]))) 127 § 4. Elementary spaces in A(qi(]))) 134 § 5. Compound spaces in A(q>(])))

Appendix

141

Index 164

Index of symbols 166

Summary 168

Samenvatting 169

For a complex Hilbert space

X,

the so-called symmetrie Fock space F (X) is defined as the direct sum of n-fold symmetrie Rilbert tensor

sym

products of X. It is a topological completion of the symmetrie tensor algebra of

X.

This Fock space, also called exponential Hilbert space, is frequently used in quantum field theory by theoretical physicists and can be represented as a functionalHilbert space S(X) with reproducing kemel K(x,y) = exp(x,y)X, x,y €

X.

The Hilbert space S(X) consists of analytic functions F on

X

with growth behaviour

2

IF(x) 1 ~ _{llFll 8 (X) exp(illxllx) ,} x E X •

In a distributional approach of quantum field theory a symmetrie Fock space construction has to be developed for locally convex spaces

V,,

e.g. test function spaces or distribution spaces, more general than Hilbert spaces. In this general setting, it is not clear in which way the symmetrie tensor algebra of

V

can be topologized or completed, a difficulty that arises already at the level of the n-fold symmetrie algebraic tensor products.

In the present monograph, these topological problems are solved rather elegantly, constructing a 'symmetrie Fock space' of

V

which consists of analytic functions on the (strong) dual

V'

of

V.

In this context we observe that also the functional Hilbert space S(X1 ) represents the symmetrie Fock space F _sym (X).

This thesis deals with a class of locally convex spaces, which admit such symmetrie Fock space construction leading to locally convex spaces of the same class. Our class consists of spaces which are inductive limits of Hilbert spaces, projective limits of semi-inner product spaces or both at once. The construction is carried out only for sequence spaces and as a result we obtain locally convex spaces of analytic func-tions on sequence spaces. However, the concepts can be easily'.adapted for more general locally convex spaces.

The undèrlying treatise is based on an amalgamation of ideas from - Aronszajn's reproducing kernel theory,

Bangmann's construction of Hilbert spaces of analytic functions, which are repiesentations of the symmetrie Fock spaces F (ll:q) and

sym F sym (R,2).

general theory of analytic functions on locally convex spaces as described by Nachbin and Dineen,

discussions on inductive/projective limits of (semi-) inner product spaces by Van Eijndhoven, pe Graaf and Kruszynski.

1

Let us sketch our approach with an example. The spaces we start from in this example are the so-called analyticity spaces and trajectory spaces introduced in [G]. Spaces of these types be long to our class and have been a major source of inspiration.

For a positive self-adjoint operator A in X, the continuous chain of Hilbert spaces Xt, t E JR, is defined as follows. For t ~ O, Xt is the space e-tA(X) endowed with the inner product (x,y)t

=

(etAx, etAy)X. For t < O, Xt is the completion of X with respect to the inner product

(x,y)t

=

(éAx, etAy)X, There is a natural duality between the spaces Xt and X_t. The set {Xt 1 t > O} is an inductive system with the

ana-lyticity space

SX,A

as its inductive limit. The set {Xt 1 t < O} is a projective system witb the trajectory space TX,A as its projective limit. The spaces

SX,A

and TX,A are in strong duality.

In [EG 2] the spaces SFs (X),H and TFs (X),H are introduced for Ha suitable self-adjoint opr:ator

associat~

with A. These spaces are topological completions of the symmetrie tensor algebra related to

SX,A

and

TX

A' respectively .

•

Besides, the analyticity space SFs (X),H and the trajectory space TFsym(X),H admit a representation: spaces of analytic functions. :_ndeed, in [Ma 1 ] the s paces SS (X),

H

and T S (X) ,

H

are in troduced, where

H

denotes a positive self-adjoint operator in S(X) related to A. It bas been proved tbat the space SS(X),H consists of analytic functions on the strong dual TX,A of SX,A and tbat the space Ts(X),H consists of analytic functions on the strong dual SX,A of TX,A'

In this. thesis we replace SX,A and TX,A by more general inductive limits

V and projective limits W, which are in duality, and we construct spaces of analytic functions on Wand

V,

respectively, similar to SS(X),H and TS(X)

,H·

In the remaining part of this introduction we present a summary of each chapter.

Chapter I contains prerequisites of the present monograph. We recall results on locally convex spaces, in particular, results on inductive and projective limits and results on analytic functions on locally convex spaces. As an illustration serves the space of analytic functions on the space of finite sequences. This function space is an important tool in the aforementioned constructions.

Chapter II is devoted to Aronszajn's theory on reproducing kemels and functional Hilbert spaces, We summarize results of Aronszajn's :theory and represent the symmetrie Fock space F (X) as a functional Hilbert

sym

space S(X). In a separate section we pay attention to examples of functional Hilbert spaces, which are interesting for their own sake. These examples are not relevant for the following chapters.

Chapter III is an exposition on inductive limits of Hilbert spaces and projective limits of semi-inner product spaces. It contains a reformula-tion and a simplificareformula-tion of the theory as developed by Van Eijndhoven, De Graaf and Kruszyóski. Starting from a Köthe sequence set and a countable collection of Hilbert spaces, we construct an inductive limit of Hilbert spaces and a projective limit of semi-inner product spaces and describe their duality. Topological prope.rties of these inductive/ projective limits are put in correspondence with properties of the Köthe set and the collection of Hilbert spaces .• In particular, certain Köthe sets result in locally convex spaces, which are both inductive limits of Hilbert spaces and projective limits of Hilbert spaces.

Chapters II and III contain the main tools for the final chapter, Chap-ter IV, and can be read independently.

In Chapter IV, for Köthe sets

g,

we introduce our sequence spaces

inductive limits Gind [g] arise from weighted R.2-spaces R.2 [a;ID], fixed

by nonnegative sequences ä on countable sets ]). The projective limits

+ +

G • [g] arise from semi-inner product spaces R.

2[a;ID]. Here R.2[a;ID] prOJ

denotes the Köthe dual of R.

2[a;ID].

Applying reproducing kemel theory, for nonnegative sequences a on 1D we introduce our elementary function spaces

F.

d[a] and

F

.[a]. The

in proJ

functions in Find [a] are analytic on R.;[a;1D] whereas the functions in F .[a] are analytic on R.

2[a;ID]. Fora Köthe set g the collection prOJ

{F. _in d[a] 1 a E g} turns out to be an inductive system and the collection {F .[a] 1 a € g} a projective system. Our compound spaces

F.

d[g] and

prOJ in

F _prOJ .[g] are defined to be the corresponding inductive and projective limits, The space

F.

d[g] consists of analytic functions on G .[g] and

i.n · proJ

yields a description of the symmetrie Fock space of G. _in d[g], Likewise, the space _. F _proJ .[g] is a space of analytic functions on G. d[g] and _in yields a description of the symmetrie Fock space of G _proJ .[g]. We charac terize the func tions in F • . d [g] and F . [g] wi th a growth

in proJ

estimate and we characterize them also in terms of the coefficients in their monomial expansions. The latter characterization leads to sequence spaces G. d[up[g]] and G .[up[g)], homeomorphic to F. d[g] and

in proJ i.n

F .[g], respectively, with up[g] a Köthe set on multi-indices. prOJ

Studying these sequence spaces, we obtain topological properties of the corresponding compound spaces.

As an illustration of the theory we represent suitable infinite dimen-sional Heisenberg groups as groups of linear hcmeomorphisms on the spaces F. d[g] and F • [g].

In this chapter we present the prerequisites of the entire monograph.

§ 1. Locally convex spaces

In this section we present definitions and results on locally convex spaces. Sources are the well-known textbooks [Yo], [Con] and [Sch].

(1.1) A aomplex veator spaae is a set V with two algebraic operations,

addition and scalar multiplication, which satisfy:

(V,+) is an abelian group.

- The scalar multiplication is a mapping from C x V into V for which for all À,µ € C and v,w €

V

À(v+w) • Àv + Àw (À+ µ)v Àv + µv

(Àµ)v À(µv)

1v

=

v •

In general, we omit the adjective 'complex' in the expression 'a com-plex vector space'.

A t(lpological veator spaae is a vector space V endowed with a topology,

such that the mappings

(v,w)i+v+w, (v,w) E V x V

and

are continuous with respect to the product topology. Hence for an open set W in a topological vector space V the sets ÀW and v + W are open for

all À E

t

and v E V.

(1.2) A set W in a vector space V is

aonve;i; if \J

Vv,wEW vÀ,0<À<1 Àv + (1-/,)w E W

balan,..ed if .., u

" vvEW vÀ€(:, 1À1 ;l;l ÀV € W

absorbing if VvE V 3À>O : Àv E W •

(1.3) A locally aonvex apaae is a Hausdorff topological vector space, such that the topology is locaUy aonve~, i.e. each neighbourhood of o contains a convex, balanced and open set. We abbreviate 'locally convex space' and 'locally convex topology' by 'L.C. space' and 'L.C. topology'.

A semino:rrn p on a vector space V is a function

p:

V + IR+, such that for

all À €

t

and v,w €

V

p(v + w) :i p(v) + p(w) p( v)

1.1. Lemma

Let V be a (topological) vector space. Let W be a convex, balanced and absorbing (open) subset of

V.

The Minkowsky functional or gauge

kw•

defined by

1

-1

Itw(v)

=

inf {À >

0

À v €

W} ,

v €

v

is a (continuous) seminorm on

V.

1.2. Lemma ..

Let V be a (topological) vector space. Let p be a (continuous) seminorm on V.

The set

Up=

{v €

V

j p(v) < 1} is a convex, balanced and absorbing (open) subset of

V.

The previous lemmas are the key to the main result in this section, namely the relation between L.C. topologies and collections of seminorms.

(1.4)

A collection of seminorms

Pr

is

{py

j

y E r} on a vector space V

sepaPating if for each v E

V,

v f. o, there exists y E

r

such

that py(v) f. 0 ,

di~eated if for each y₁,y₂ € r there exists y Er such that for all v E V: p (v) ::î p (v) and p (v) ::î py(v).

Y1 y Yz

(1.5) All subsets Win

V,

satisfying the condition:

For each w E

W

there exist .a finite set E c:

r

and s > 0 such that w +

n su

c:

w,

yEE Py

establish a topology, T(V,pr)·

Let T denote a topology in a vector space

V.

The collection of semi-norms

Pr

on V gene~ates T if T

=

T(V,pr>·

Here follows the main result in this section.

1 . 3. Theorem

Let V denote a vector space endowed with a topology T.

Then V is a L.C. space iff T is generated by a separating collection of seminorms.

Let

Pr

and qfi denote two collections of seminorms on a vector space

V.

The collections

Pr

and qfi are equivalent if

Pr

and qó generate the same topology; to state it differently, the seminorms Py~ y E r, are contin-uous with respect to the topology T(V,qfi) and the seminorms q

6,

o

E ó,

are continuous with respect to the topology T(V,pr).

The following result is useful.

1. 4. Lemma

Let

Pr

and qfi denote two directed collections of seminorms on a vector space

V.

and

(1.6) A subset W of a L.C. space V is bounded if for each neighbour-hood

U

of o there exists À > 0 such that À-1

w

c

U.

1.

s.

Lemma

Let

W

denote a subset of a L.C. space

V.

Then the set

W

is bounded if f for all continuous seminonns p on

V

sup {p(v) 1 v E W} < 00 •

In a L.C. space all compact subsets are closed and bounded.

The space of all continuous linear operators from a L.C. space V into a L.C. space

W

will be denoted by

B(V,W).

We write

B(V)

instead of

B(V,V).

1.6. Lemma

Let V and W denote L.C. spaces with generating directed collections of seminorms

Pr

on

V

and q/:J. on

W,

respectively. Let

A

denote a linear operator from

V

into

W.

Then

A E B(V,W)

iff

1. 7. Corollary

Let V and W denote L.C. spaces. Let B be a bounded subset of V and let A E

B(V ,W).

Then A(B) is a bounded subset in

W.

(1.7) The

du.at

of a L.C. space Vis the vector space of all continuous linear functionals on

V

and is denoted by

V'.

For each bounded subset W of V we define the seminorm -OW on V' by

The ûJeak dual of a L.C. space V is the space Vf endowed with the topol-ogy

T(V', {-OW

1

Wis

a finite subset of V}).

The weak dual of

V

is denoted by

V&·

The strong dual of a L.C. space V is the space V' endowed with the

topology T(V',

{-OW

1

W

is a bounded subset of V}). By

VS

we denote the strong dual of

V.

Weak duals and strong duals of L.C. spaces are L.C. spaces.

(1. 8} For each v E

V

we define the linear functional .ev on _v

V'

by ev _v (i)

=

i(v) , i E V'

The L.C. space V is reflexive if the mapping v i+ evv is a homeomorphism

from

V

onto (V$)e•

(1.9) A Cauchy net (vi)iEJ in a L.C. space V ie. 'a mapping j i+ vj from

a directed set J into the space

V

such that for each neighbourhood

W

of o in V:

3.EJ V.EJ '<" v. - v. E W •

l J 'l.~J l J

A Cauchy net (vi)iEJ in

V

has a limit v E

V

if for each neighbourhood

W

of o:

v-v.EW. J

The L.C. space

V

is aonrplete if each Cauchy net has a limit.

(1.10) The L.C. space Vis semi-MonteZ if each closed and bounded subset . of V is compact. The semi-Montel space

v·

is MonteZ i f V is reflexive.

( 1. 11) A c losed, convex, absorbing and balanced subset W of a L.C. space

V is a barrel. The L.C. space V is barreZed if each barrel in V is a neighbourhood of o in

V.

(1.12) The L.C. space V is bornologicaZ if every convex balanced subsèt

(1.13) A linear operator S from a Banach space

V

into a Banach space

W

is called nuaZear if there exist bounded sequences (v~)nEIN in

VS

and (wn)nEIN in W and a sequence C € t_{1 (IN) with nonnegative entries such} that

S(u)

=

l

c(n)v' (u)w ,

nElN n n u €

v .

If

V

and

W

are Hilbert spaces, then the operator S is nuclear iff S is a trace class operator.

Let V denote a vector space, p a seminorm on V and Np" { u € V 1 p( u) " 0}.

The normed linear space V is the quotient space V/Np with norm

p

de-"' p À

fined by p(v + N ) " p(v), v E V. By V we denote a completion of Vp and

p À p

we consider

Vp

as a subspace of

Vp.

Let

q

denote another seminorm on

V

with VvEV : p(v) :îi q(v),

The.n VvEV: v + N cv+ N and p(v+N):;; q(v+N ).

q ~ - p q

The canonical mapping jq,p: Vq +Vp is the uniquely determined continu-ous mapping, which satisfies jq,p(v+ Nq)

=

v + Np, v € V.

A L.C. space

V

is nuclear if for each continuous seminorm p there exists a continuous seminorm q such that

VvEV : p(v) :;; q(v) and

A A

the canonical mapping j _p,q

V

_q +

V

_p is nuclear.

§ 2. Inductive and projective limits

In Chapter III we study dual pairs of L.C. spaces consisting of an inductive li111it and of_ a projective limit. These inductive limits and projective limits originate from directed collections of (semi-) inner product spaces.

In this section we mention some generalities on inductive/projective limits. For further references see [Sch], Chapter II, Sections 5 and 6 and [Con], Chapter IV, Section 5.

(2.1) Let

V

and

W

denote two topological vector spaces with

V

c

W.

If the canonical injection is continuous, we express this by writing

V c;.. W.

2. 1. Deflnition

Let A denote a directed index set.

A collection of vector spaces· {Va

!

o: E A}, each endowed with a L.C. topology, is an inductive system if for each

a,6

€

A,

with o: $

S,

Vac;..Ve.

2. 2. Deflnition

Let

{Va

1

a

€

A}

denote

an

inductive system,

The inductive limit, lim ind V , is the vector space U V,., endowed with

o: E A 0: aEA "

the finest L.C. topology, such that Va c U V for each 13 E A. This "" aEA a

topology is called the inductive limit topology.

A neighbourhood basis B of o in lim ind V is formed by all convex o o: E A o:

balanced set U c U V~ such that for all ex E A the set U

n

V is open

o:EA" · a

in

va.

A criterium for continuity of mappings on inductive limits reads:

2. 3. Theorem

Let {Va 1 a E A} denote an inductive system. Let F denote a linear map-ping (seminorm) from lim ind V into a topological vector space (IJ (into

o: E A

.IR+).

Then F is continuous iff for each

S

E A the linear mapping (seminorm) FIVa is continuous on v₈•

Proof:

Cf. [Con], Section IV.5.

The second part of this section deals with projective limits.

2.11. Deflnitioh

Let A denote a directed index set.

A collection of vector spaces

{Va

1 o: E A}, each endowed with a L.C. topology, is a p~ojective system if for each o:,S E A, with a $ $, Vac;..Va.

8

2.5. Definition

Let {Va 1 a E A} denote a projective system.

The projeative Zimit, lim proj V , is the vector space

I Preliminaries

n

v

endowed a€A a

a E A et

with the coarsest topology, such that

n

v

c;_

va

etEA et for each 8 E A. This

topology is called the projeative limit topoZogy.

We remark that for the of v E

n

V is given

etEA a

projective limit topology a neigbbourhood basis by all intersections fl

F

1 (U ) , where U,.... is a

aEE a a "' neighbourhood of v in

va

and where E is a f inite subset of A. The next results are not very surprising.

2.6. Lemma

Let {Vet 1 a E A} denote a projective system where each Va is endowed

with topology T(Va•Pra). Let r • a~A ra and let Pr

=

{py restricted to fl V j y E r}. Then

a.EA a

a. The topology of lim proj V,,,,, is equal to T(

n

V,...., Pr).

aEA "' aEA....,

b. The space lim proj Va is a L.C. space iff the collection Pr is

, et E A ·

separating. Proof:

See [Sch], Chapter II, Section 5.

2. 7. Theorem

Bet {Va. 1 a E A} denote a projective system. Let F denote a linear

map-ping from a topological vector space

W

into lim proj

V •

a. E A a

Then F is continuous iff f or each a. E A the mapping F is continuous from W into V ó'.•

Proof:

See [Sch], Chapter II, Section 5.

As is well known, eacb complete L.C. space is a projective limit of Banach spaces.

In the next section we give two classica! examples, one of an inductive limit and one of a projective limit. See Lemma.3.t.

From Lemma 2.6 it follows that not every projective limit is Hausdorff. In general, it is hard to check whether an inductive limit is Hausdorff.

IJ

§ 3. Sequences and sequence sets

Throughout this section, Il denotes a fixed, countable set. Functions from Il into

t

will be called sequences (labeled by Il) and tbey are denoted by

a,b, ".,

etc. By w(ll) we denote the set of all sequences

labeled by ll.

For each subset JE of

U

the sequence

Xm

is defined by

j

e:

JE

j t E

So

Xm

is the characteristic function of the subset

E.

In particular, we errploy the notation

n

=

Xn• 0

=

x

₀

.

Fix u € w(Il). The suppor-t of u, which we denote by ll[U], is the set {j € Il

i

u(j)

r

O}. lts complement is denoted by llc[u], viz.

Xc(U]

=

{j E U

!

U(j)

=

O} •

Correspondingly we have the functions

nu

and Ot1•' defined by and

Hence for all j

e:

l[

We mention the following operations on sequences: Let

u, v

E w(lI), À

e: t:.

Add.ition. The sequence u+ v € w(X) is defined by

[u + v](j) = u(j) + v(j) , j

e:

Il •

Saa'lar multipliaation. The sequence ÀU

e:

w(X) is defined by

[ÀU](j)

=

À[U(j)) ,

je:x.

Pl'odwt. The sequence u •

v

€ w(lI) is defined by

10 I Preliminaries

Thus w(ll) becomes a commutative algebra.

Tensor product. Let w E w(Il) with

JI

a countable set. The sequence

u © w E w(n: x

JI)

is def ined by

u 0 w(i,j) = u(i)w(j) , (i,j) E ][ x JI •

Absolute value. The sequence lul E w(JI) is defined by

lul(j)

=

lu(j)I j E ll •

Pseudo-inverse. The sequence u-1 € w(ll) is defined by

l

[u( ')

1-1

-1 J u (j) = 0 j E l l . . . -1

The sequence u is called the pseudo-inverse of

u.

-1 -1

We remark that u • u

=

u • u

=

11 u.

Let p, o denote subsets of w(JI) and let T denote a subset of w(n:).

We employ the following notations: u • p = {u • r r E p} •

p•o•{r•s rEp,sEo},

pl!h={r©t r E p , t E T } , etc.

We introduce the following linear subspaces of w(JI):

(3.1) The space of finite sequences q:>(lI), defined by q:>(lI) =.{v € w(l[) 1 #(lI:[v]) < oo} •

(3.2) The space of null sequences c

0(:n:), defined by c

0(1I) = {vE w(l[)

i

Ve:>O: #({j EU 1 IV(j)t > e:}) <co}.

(3.3) The Banach space of all bounded sequences ~₀₀(ll), defined by ~

₀₀

(]() = {v € w(U) I sup({ lv(j) 1 1 j € ](}) < co}

with norm lvl₀₀

=

sup({V(j)

I

j € n:}).

i\

Il

I'

(3. 4) For p € 1N the Banach spaces R,p (lI), defined by

R, (][) -

{v

€ w(lI)

1 I

lv(j)

1P

< <XI}

p jElI

Evidently, the space t

2(lI) is a Hilbert space and we denote the inner product by (•,•)

2•

We now endow the spaces q>(Il) and w(lr) with suitable L.C. topologies. To this end we introduce the following notion:

(3.5) A sequence (lrq)qElN is called an e~haU8tion of lI, if it satis* fies:

u

lI =lI q€1N q

Each enumeration of ][ induces an exhaustion.

Assume that we have an exhaustion. Let q E lN.

We define the seminorm p on w(lI) by p (X) = 1 Xu • x 1

2 •

q q q

The q-dimensional Hilbert space q> (lI) is the space x • q>(:O:) with

q liq

norm

x

1+ lxlI • xl

2•

q

The topological vector space w (:0:) is the space w(lI) with the topology q

generated by the seminorm

p •

_q

The collect ion {<Pq (:0:) 1 q E JN} is an inductive system and

u

<P (:0:) = (fl(lI) •

q€1N q

The collection {w (lI.) 1 q € lN} is a projective system and

q

n

w (lI)

=

w(:n:) •

In the sequel, by ip(JI) and w(:n:) we mean the L.C. spaces lim ind <P (:n:)

q EIN q

and lim proj

w

(lI), respectively.

q E IN q

For u € w(lI) respectively ip(JI) we define the seminorm qu on ip(JI) respectively w(lI) by

qu (x) = 1 x • u 1

1 , x E qi(JI) respectively w(Il) •

The next lemma establishes that the topologies on qi(:lI) and w(:n:) are Hausdorff and do not depend on the particular choice of the exhaustion,

3.1. Lemma

a. The inductive limit topology on ip(lI) is equal to

T (i.p(lI), {qu 1 u E w(JI)}) •

b. The projective limit topology on w(E) is equal to

T(w(E), {qa 1 a € !.P (1[) }) •

Proof:

a, Let U E w(JI) and let q .E IN. Then

Hence the seminorm qui(.!) (Il) is continuous. By Theorem 2.3 it follows that qu is a continuous qseminorm on tp(ll)

=

lim ind tp (ll).

Conversely, let p denote a continuous seminorin E

~

l i : ind <P (JI) and

+( ) • (') ( ) qEIN q

let u E w :n: be defl.ned by u J

=

p X{j} , j E ll.

For all x E (.!)(Il) we have

p(x) :>

L

p(X{·}·>lxO>I = qu(x).

jEll J

Lenu:na 1.4 yields statement a.

b. By Lemma 2.6 the projective limit topology on w(lI) is equal to T(w(JI), {pq 1 q € IN}). With the aid of Lemma 1.4 we can prove that

The spaces tp(ll) and w(ll) are Montel spaces. This result is a conse-quence of the theory stated in Chapter III as is the case with the following characterizations.

(3.6) Let

W

denote a subset of q>(ll).

The set W is compact iff there exists q € IN such that W c: q> (:U:)

q and W is compact in q>q (ll).

(3.7) Let

W

denote a subset of w(ll).

The set W is compact iff for each j € 1I the set {u(j) 1 u E W} is compact in

t.

We introduce the set of multi-indices labeled by E.

3. 2. Defini tion

Let

n

denote a countable set.

The set of multi-indices l1(Il) c:q>(Il) is the set of all sequence from

n:

into IN+ (= 1N U {O}) with a finite number of non-zero entries, i.e.

l1(JI)

=

{s: lI + lN+ 1 #(ll [s]) < oo} •

We call l1(ll.) the multi-index set labe'led by IL For n E IN+ we also define

(3.8) For elements of l1(ll) we introduce the following corresponding multi-index notations.

For each

s

E l1(ll) and a E w(Il): s 1

=

n

s (j) 1 • jEll a5 ₌

_n

_{[a(j)]s(j) •} jEJI [s] In particular, we have

a~

=

1. We remark that 11({1, ••• ,q}) =

m1.

§ 4. Analytic functions

In this section we introduce the concept of analytic function on an open subset W of any L.C. space according to S. Dineen [Di 1] and present some of its consequences. We study two examples: The space of analytic functions on q>(ID) and the space of analytic functions on w (ID). (ID is a countable set.) But we first have to deal with the special case of analytic functions of a finite number of complex variables.

Following L. Hörmander in [Hö], we introduce the notion of analytic function on an open subset in ~q by means of the Cauchy-Riemann equa-tions.

Let

W

denote an open set in

~q.

By Ck(W) we mean the space of k times continuously differentiable complex valued funétions on

W

where

k

=

oo,1,2, •••• To each function F €

c

1(W) we associate the differential form dF of the function F.

To this end, we identify

fq

with 1R2q, viz.

z

=

x + iy '

Let F € C 1 (W). Considering F as a function of 2q real variables it

k . ()F d óF 1 ,,. • <

ma es sense to wnte ax(j) an

äYITT , "

J = q. The expression

q óF ~

oF

l

ax(3•) dx(j) + t.. ay(3') dy(j)

j=1 j=1

is called the differentiaZ fo1'm of F and is denoted dy dF. Put

dF 1 êF i óF

äZGî

~

₂

axGT

~

₂

äYITT ,

dz(j) • dx(j) + idy(j) and

dZ(j) = dX(j) - idy(j) •

Then

dF =

!

a:CJ•)

dZ(j) +

!

~

dz(j) ,

With q and aF

=

I

~

dï'îJ) j=1

azur

we have dF

=

()F + ä'F • 4. 1. Definition

A function F E

c

1(W) is analytic if äF • O on W, i.e. F satisfies the Cauchy-Riemann equations on

W

By A(W) we denote the vector space of all analytic functions on

W.

First we present some results related to Càuchy integrals. For u E 4:q and r E IRq with r(j) > 0 we denote the polydisc

lw(j)- u(j) 1 ~ r(j), 1 ~ j ~ q}

by n B(u,r), its interior by n B(u,r), .and its distinguished boundary _{q q}

by n aB(u,r).

q

lw(j) - u(j) 1 = r(j)}

4.2. Lemma ([Hö), Theorem 2.2.1)

Let F be a continuous complex valued functión on an open subset

W

C:: tq. Then the funct!on Fis analytic on

W

iff for all polydiscs nqB(u,r)

c:W

and all V E

n

B(u,r) q

F(V) (27Ti) -q

n élB(u,r)

q

Here we use the multi-index. notation:

q (w-v)11

=

n

(w(j)-v(j))1 j=1 See also (3. 8). and d W

=

dW(1) " . dW(q) • q

4. 3. Corollary ([Hö], Corollary 2.2.2) Let F E A(W).

Then the function F belongs to c"'(W) and all derivatives of F are ele-ments of A(W).

Let F E A(W). By a.F we mean the partial derivative of F, corresponding h ·th . blJ . " (lp 1

to t e J varia e, 1.e. "j F

=

oZ(j) , ;;;; j ;;;; q.

Fors E

IN~

we denote the partial derivative [

ri

()~(j)]

F by

a

5F.

j=1 J 4. 4. Corollary

Let F E A(W) and let s € IN;.

Q

Let n B(u,r) c W.

q Then for all V E TI B(u,r)

q

4. 5. Corollary

f

Let F E A(W) and let nqB(u,r)

cW.

There exists µ > 0 such that for all

F(W)/(w -v)5+11 dw. q

As in the one dimensional case, analytic functions of several variables have power series expansions.

4.6. Theorem ([Hö], Theorem 2.2.6)

Let F denote a complex values function on an open set

W

c~q.

Then F belongs to A(W) iff for all nqB(u,r)

cW

and all v € TiqB(u,r)

F(V)

l

ft (()

5 FJ(U) • (V - u)5 ,

sEm;

where the series converges absolutely and unifonnly in

v

on

n

B(u,r). <! The power series in Theorem 4.6 can be replaced by a series of'homo-geneous polynom.ials.

4. 7. Definition Let F E A(W).

Then for each

u

E Wand n E lN+ we.define the n-homogeneous polynomial F [u], associated tó F, by

n

v € t:q •

4. 8. Definition

Let F denote

a

complex valued function on an open subset

W

c: Çq. Then F is ray-analytic on

W

if for each

u

E

W

and

v

€ t:q the function

À + F(u + ÀV) is analytic in a neighbourhood of 0 in C.

4.9. Lemma Let F € A(W).

Then F is ray-analytic on

W

and for each U €

W

and V €

Cq

there exists a neighbourhood

U

of 0 in 4: such that

00

F(u+ ÀV) •

l

Àn Fn[u](v) , n-0

À€

u

where the series is absolutely convergent in À on

U.

The following lemma is a consequence of Lemma 4.2.

4.10. Lemma

Let F denote a continuous and ray-analytic function on an open subset

W

in (:q.

Then F is analytic on

W.

The continuity condition in the previous lemma is superfluous •. Indeed, Hartogs' theorem yields the converse of Lemma 4.9. So it yields a remarkable characterization of analytic functions in q variables.

4.11. Theorem {Hartogs' theorem, [HöJ, Theorem 2.2.8)

Let F denote a ray-analytic function on an open subset

W

in (:q. Then the function F is analytic on

W.

Next we extend the notion of analyticity to functions defined on an open subset of a L.C. space.

4. 12. Definition

Let F denote a complex valued function on an open subset

W

of a L.C. space V.

The function F is ray-analytia on W if for all u

E

W and v

E

V the function À 1+ F(u+ ÀV) is analytic in a neighbourhood of 0 in

c.

The function F is analytia on W if F is continuous on W and ray-analytic on

W.

By A(W) we denote the space of all analytic functions on W.

It follows from Hartogs1 _{theorem that a function F is analytic on a}

L.C. space

V

iff F is continuous on

V

and ray~analytic on each finite dimensional subspace of

V.

For the following results we refer to Dineen [Di 1], Chapter 2, Sections 1 and 2.

First we introduce n-homogeneous functions associated to a ray-analytic function. An n-homogeneous function G on a vector space V is a function which satisfies

G(ÀV)

=

Àn G(v) • ÀEC, VEV.

4. 1 3. Deflnition

Let F denote a ray-analytic function on an open subset Win a,L.C. space

V.

For u E W and n E lN+ we define the n-homogeneous function Fn[u]: V-+ C by

F [u]Cv) • lim

-t---

J

F(u+ ÀV)/Àn+1 dÀ •

n p+O ~1

IÀl•p

v €

v .

We give two characterizations of the concept of analyticity.

4.14. Lemma

Let F denote a ray-analytic function on an open subset

W

in a L.C. space

V.

Then

v 1+

Iri'=0

Fn[uJ(v) defines a continuous function in a neighbourhood

of o in V.

b. The function F belongs to A(W) iff F is locally bounded on W, i.e. for each u E

W

there exists a neighbourhood

Ou

of u in

W

such that F(Ou) is a bounded set.

Proof:

See [Di 1], Section 2.2.

We give an example, interesting in itself, but also important as a tool in Chapter II. Let H denote a Hilbert space with inner product (•,•)H and let y E

H.

The function F on

H

is defined by F(X)

=

exp(x,y)H, x E

H.

The function F is analytic on

H.

The n-homogeneous functions . Fn[u], u E

H,

are v t+ (v,y)n exp(u,y)H, v E

H.

We now come to the two examples mentioned _in the introduction of this chapter, viz. the analytic functions on q>(lD) and on w(lD).

First we deal with q>(lD). As we have seen, the vector spaoe q>(lD) is a union of finite dimensional vector spaces with inductive limit topology. We show that this topological property of qi(lD) implies that each ray-analytic function on q>(lD) is continuous on q:i(lD). Thus we arrive at an extension of Hartogs' theorem, for which we present a detailed proof, because we could not find a reference.

IJ. 1

s.

Theo rem

Let F be ray-analytic on i.p(lD). Then F is analytic on q>(lD). Proof:

Let (ID q) qElN be an exhaustion of ID and let !Pq (ID) be the fini te dimen-sional vector space xlD • q>(lD) with norm x 1+ lxlD • xl

2• Then

q q

q>(ID) lim ind i.p (lD) q E lN q

Fix q E lN. Since tpq (lD) is a fini te dimensional vector space, Hartogs' theorem implies that the restriction FllA (lD) is analytic on <Pq(ID). Hence FjlA (ID) is continuous on \Pq(ID). ihe only failing argument in the proofqis the continuity of F on 1P(lD). For this we prove the

remarkable fa.et that each function on q>(ID), continuous on every <Pq (ID),

is continuous on q>(ID) itself.

Indeed, let u E q>(ID) and let e: > O. Then u E IP (ID) for some p E IN.

p

Hence there exists 6 Em., 0 < 6 < 1/2P, such that

p p

"'xE (ID) : lx- u 1₁ < o • IF(x) - F(u) 1 < e:/zP •

(j)p p

Let q >p. The set C

=

{v E (j) (ID) J lv-ul

1 1} is compact in lP (ID).

q q q

Hence there exists 6 , 0 < 6 < 1/2q, such that

q q

"'xyEC :lx-y11<éi •IF(x)-F(y)l<r;,/2q.

• q q We define a E w(ID) by

~

110 , a(j) = P 1 / éi _q ' . E ID J p j E ID '-ID 1 , q > p q

q-and the continuous seminorm qa on q>(ID) by

Let v E l(lq (ID) for some q E lN with qa (v - U) < 1. We have

q

IF(V) - F<u> 1 :il

r

IF<v. xID ) - F(V. xID >1 +

k=p+1 k k-1 + JF(v•xID )-F(u)I p Since 1 q qa (v - U) = ₀ J (V. - U)XID J 1 +

l

J V • XID , 10 11 < 1 , p p k=p+1 k k-1 for p < k :il q 1 v •

x

₁₀ - u 1 1 <

o

p p v • Xm E ck k 1

v •

x

_{10 -}

v • Xm

₁₁ <

ok •

k k-1

Hence

q

IF(v) - F(u)I

~

L

E/2k + E/2P

~

s .

k-p+1

4. 16. Corollary

Let F denote a complex valued function on q>(ID).

The function F belongs to A(ll)(ID)) iff for all finite dimensional sub-spaces W of q>(ID) the function

Flw

is analytic on W.

As all spaces A(V), the space A(!.l>(ID)) is an algebra, because for F ,G € A(q>(ID)) the function F • G is analytic on q>(ID).

We need a pairing between q>(ID) and w(ID).

4.17. Definition

The pairing <•,-> between q:i(ID) and w(ID) is defined by <x ,y> • }'. x(j

>YGT •

j€ID

where x € q>(ID), y € w(ID) or x € w(ID), y € q:i(ID).

We define the following operations on functions F in A(<,p(ID)).

(4.1) The translations Ta, a € q>(ID), defined by

[Ta F](x)

=

F(a+ x) , X € q>(ID) • (4.2) The shifts

Rb,

b € w(ID), defined by

[Rb F](X)

=

e<x,b>

F(X) •

x

€ q:>(ID) • (4.3) The dilatations 0b, b €.w(ID), defined by

(4.4)

[0b F](x) = F(b • X) , X € ip(ID) •

The multipliers

Q.,

j €ID, defined by J

[Qj F] (X) = <x,e/. F(X) • x € Q)(ID) • where e . = X{ • } , j € ID •

J ' J

(4.5) The differential operators

a.,

j E lD, defined by J

[o

3. F](x) = lim _;\....O [F(x + ;\e.) - F(x) _J ]/À ,

x

E (j)(lD)

(4.6) We recall that lM(lD) denotes the set of multi-indices labeled by lD. Cf. Definition 3.2.

The differential operators 35,

s

E lM(lD), defined by

a

5 _F=

_n

_[a~(j)]F.

jElD J

Theorem 4.15 and Corollary 4.16 imply that these operations are indeed linear mappings from A(q:i(lD)) into A(<P(lD)).

As in Cq for each u E t:p(lD), each fini te subset 1F in lD and each sequence r on lF with positive entries, we denote the polydisc

{w E <P(lD)

l

lD[w] clF, vjEJF: lw(j)- u(j) 1 ;:;;. r(j)}

by nF B(u,r). Its distinguished boundary

{w E tp(lD)

l

lD[w] c:JF, vjEJF: lw(j)-u(j)J r(j)}

is denoted by OF <lB(u,r).

We have the following Cauchy integral representation for the partial derivatives.

4.18. Lemma

Let F E A(tp(lD)), u E q>(lD) and S E lM(lD).

For fini te subsets F of lD w.ith lD[u], lD[S] c F we have

(27Ti)-#(JF) where dF w means

n

dW (j) . jElF Proof:

f

F(W)/(w - U) s+x lF dF W nF ClB(u,r)

The proof is based on Lemma 4.2 and Corollary 4.4 for a sufficiently large fini te dimensional subspace of q>(ID). D

As in the case of q c0111.plex variables, analytic functions on qi(lD) have a power series expansion.

IJ. 19. Theorem

Let F E A(qi(lD)). Then

vxEqi(lD) : F(x) = }'.

-fï

[as F](o)xs SElM(lD)

where the series converges absolutely and uniformly in

x

on compact sets in q>(lD) ~

Proof:

Let (ID ) ElN denote an exhaustion of lD and let q> (lD)

=

xID • <P(ID).

q q q q

The function F\ (lD) is analytic and can be expanded in a power series. IPq .

Hence

"'xE

(ID) : F(x) "

l

J...

[as F]

(o)xs •

<Pq SE:M(ID),ID[S]cID SI

q

where the series converges absolutely and uniformly in x on compact sets in q>q(ID). See Theorem 4.6.

The proof is canpleted because the canpact sets in q>(lD) are always canpact sets in some q> (ID). Cf. (3. 6). c

q

Although q>(ID)

=

U qi (ID), it is not true that each function qElN q

F € A(q>(lD)) depends only on a finite number of variables; so it is not true that F =

BxF

F for some finite subset lF of ID. As an example we mention the function G: X .+ <x,a>, x E <P(lD), where ID[a] is infinite.

The function G is analytic on q>(ID) and for each finite subset lF of ID G

F

9xF G.

We finish the discussion of A(IP(lD)) with some remarks on the transla-tions T a and the shifts Rb.

(4. 7) Let G{q>,w}(lD) denote the group q>(ID) x w(ID) x

t

with product operation

U(a,b,c): A(<P(ID))-i- A(q>(ID)) is defined by

, U(a,b,c)F

é

Rb T a F F E A(q>(ID)) We have

The group G{q>,w}(ID) can be regarded as a generalization of the Heisen-berg group. See the Appendix to Chapter IV. The mapping U is a represen-tation of the group G{q>,w}(ID) in A(q>(ID)).

The second example is established by the space of analytic functions on the space w(ID). It turns out that these functions have a simple charac-terization.

4. 20. Theorem

Let F denote a ray-analytic function on w(ID).

The function F belongs to A(w(ID)) iff there exists a finite subset JF

of ID such that VX(w(ID): F(x) = F(xF• x).

Proof:

-. By Hartogs' theorem the restriction FIXlF •W(ID) is continuous and also the operator x >+ XlF • x is continuous from w(ID) into XlF • w(ID). Hence Fis continuous on w(ID).

•· Theorem 4.15 says that Flq>(ID) is analytic and Theorem 4.19 yields

v F( ) \' _!_, [as F](o)xs • XEq>(ID) : X SE:m.f(ID) S.

We prove that F depends· only on a finite nl.lllber of variables. Since F is continuous, there exist

a

E

q>(ID) and µ >

O,

such that

VWEw(ID): 1a. Wl1 < 1 " IF(W) 1 < µ •

Let s E 1M(ID). We define

r

E w+ (ID) by

r

=

a + ptla for some p >

o.

With the integral representation, given in Lemma 4.18, we ,can derive the following estimate:

Since p is arbitrary, we find that for all s € lM(ID) with ID[s ]'-ID[a)

1'

t',

[as F](o) =

o.

Put F

=

ID[a], Then we have Flqi(ID)

=

(0XF F)lqi(ID)' Since qi(ID) is

dense in w(ID), we get F =

e

F. c

Xp Also the space Á(w(ID)) is an algebra.

Further, we list some operations on the functions F in A(w(ID)}, These operations are similar to the operatións in the previous example.

(4.8) The translations Ta• a E w(ID), defined by

[Ta F](x)

=

F(a+ x) ,

x

€ w(ID) • (4.9) The shifts Rb, b E q>(ID), defined by

[Rb F](x)

=

e<x,b> F(X) • X € w(ID) •

(4.10) The dilatations eb, b E w(ID), defined by

ceb F](x) = F(b • x) , X € w(ID) • (4.11) The multipliers ,Qj' j €ID, defined by

[Q.

F](x)

=

<x,e.> F(x) ,

J J X € w(ID) •

(4.12) The differential operators aj' j €ID, defined by

[a. F](x) = lim (F(X+Àe.) - F(X))/À, X € w(ID).

J Ä-+0 J

(4.13) The differential operators as, S € lM(ID), defined by

as F ... [

n

a~

(j

>]

F • jEID J

Theorem 4. 20 implies that these operations are linear mappings from

A(w(ID)) into A(w(ID)).

4. 21. Theorem

Let F E A(w(ID)). Then

vxEw(ID) : F(x)

=

l

-&

[as F] (o)xs SElM(ID) .

where the series converges absolutely and uniformly in

x

on compact sets in w(ID).

Proof:

Because of Theorem 4.20, there exists a finite set lF clD such that for all x E w(ID) F(x)

=

l

5\ [as F](o)x 5 • SE:M(ID) ,ID[s ]cF

Let W denote a compact set in w(ID). From (3. 7) it follows that XF • W

is a compact set in the fini te dimensional vector space XlF' • W(lD). Hence the last mentioned power series converges absolutely and uniformly in X on

W.

We finish this section with a remark on the 'Heisenberg group' related with A(w(ID)), Cf. (4.7).

(4.14) Let G{w,q>}(ID) denote the group w(ID) x q>(ID) x (; with product operation

For an appropriate triple (a,b,c) the operator V(a,b,c) A(w(ID)) + A(w(ID))

is defined by

V(a,b,c)F = ec Rb Ta F , F E A(w(ID)) •

Similar to the case (4.7), V is a representation of G{w,q>}(ID) in A(w(ID)).

Introduction

Aronszajn's paper [Ar] on reproducing kemels, published in 1950, is the starting point of this chapter.

A reproducing kemel reproduces a functional Hilbert space in the fol-lowing way: Let

H

denote a functional Hilbert space, i.e. a Hilbert space consisting of complex valued functions on a set E such that for all y €Ethe evaluation functionàls F '+ F(y), F €

H,

are continuous. Their Riesz representants K , complex valued functions on E, constitute

y

the reproducing kernel Kof

H,

defined by K(x,y) • K (x). Functions y

K: E x E -+ t, thus defined, are functions of positive type on E, viz.

VR.EJN VOl..€t,x.€E, 1;;i;jH

J J

Conversely, starting with a function P of positive type on E, there exists a uniquely determined functional Hilbert space H(E,P), which admits P as its reproducing kernel.

We mention an important result of reproducing kemel theory: Given functions K1 and K2 of positive type on E, we write K1 ~ K₂ whenever

Ki -

K

1 is of positive type. I f K1 ;;; K2 then H(E,K1) c::;..H(E,K2).

In Chapter IV we will consider inductive systems ·and inductive limits. The inductive systems are composed of functional Hilbert spaces, con~

sisting of analytic functions on qi(])), the space of finite sequences.

We mention three points where reproducing kemel theory enters in the considerations of Chapter IV:

First: Reproducing kemel theory provides an elegant and transparent way to introduce functional Hilbert spaces of analytic functions on q>(JD), which are in fact representations of symmetrie Fock spaces. Cf. Section IV.4.

Second: A directed set of functions of positive type on a set E corre-sponds to an inductive system of functional Hilbert spaces on E. Cf. Theorem IV.4.4.

Third: The functions in a functional Hilbert space are characterized by a growth estimate, which only involves the reproducing kemel. This result carries over to inductive limits of functional Hilbert spaces. Cf. Theorem IV.5.4.

After this explanation on the relevance of this chapter for Chapter IV, we now sketch the contents of the present chapter.

In the first section we deal with functions of positive type. We char-acterize the functions in a functional Hilbert space and study a partial ordering and some operations in sets of functiàns of positive type. It turns out that Hilbert tensor products and direct sums of functional Hilbert spaces are again functional Hilbert spaces.

The second section deals with the n-fold Hilbert tensor product Tn(H), the n-fold synunetric Hilbert tensor products T5ym(H) and the symmetrie

n

Fock space F _sym (H) of a Hilbert space

H.

Each of these spaces admits a functional Hilbert space as its representation. The representations of T8_n ym(H) and F _sym (H), denoted by S (H) and S(H), consist of analytic _n functions on

H.

(We regard Sn(H) and S{H) as the n-fold symmetrie Hilbert tensor product of

H

and the symmetrie Fock space of

H,

respec-tively.) Guichardet's monograph [Gu] on symmetrie Hilbert spaces touches upon the same subject, but is based on a completely different approach. The third section consists öf classical examples of functional Hilbert spaces consisting of analytic and harmonie functions. Our spaces S(Cq) appear to be the Bargmann spaces

B

of order q.

q

In the appendLx we stud~ linear operators in the symmetrie Fock space S{H), such as the annihilation and creation operators of quantum field theory and differential operators of infinite order.

§ 1. Reproducing kemel theory

For a greater part the results in this section are inspired by Arons-zajn' s results [Ar] on reproducing kemels. First, we introduce func-tional Hilbert spaces, reproducing kemels and functions of postive type and we explain the mutual relations between .these notions.

1 • 1. Definition

Let

H

denote a Hilbert space consisting of complex valued functions on a set E.

The space H is called a fu:national Hilber1; spaae if for each y E E the evaluation functional F 1+ F(y), F E

H,

is continuous.

Throughout, the norm and the inner product in a Hilbert space ,ff will be denoted by ll•llH and (•,•)H' respectively. For functions L: Ex E +(:we write L for the function xi+ L(x,y).

y 1. 2. Definitlon

Let

H

denote a Hilbert space consisting of complex valued function on a set E.

A function K: E x E + (: is called the reprodwing kemel of H if K bas the following two properties:

- Vy€E : The function Ky belongs to

H.

An i111111ediate consequence of the Riesz1 _{representation theorem is the}

following

1.3. Lemma

Let

H

denote a Hilbert space of complex valued functions on a set E. Then

H

is a functional Hilbert space iff

H

has a reproducing kemel K.

Reproducing kemels of functional Hilbert spaces are uniquely deter-mined.

Let K be the reproducing kemel of a functional Hilbert space

H

on E. For all Jl. € IN and a. € (:, y. E E,

According to the following definition K is a function of positive type.

1 , 4. Definition

Let P denote a function from E x E into ~.

The function P is called a funation of positive type on E if for all

~ E 1N and a. E C, y. E E, 1 ~ j ::! ~.

J J

By PT(E) we denote the set of all functions of positive type on E.

Each reproducing kemel of a functional Hilbert space on E is a function of positive type on E. The converse is also valid.

1. 5. Theorem ([Ar], Part 1, 2( 4))

Each element P of PT(E) induces a uniquely determined functional Hilbert space on E, which admits P as its reproducing kemel.

Proof:

We give a sketch of the proof. In the linear space V(E,P)

=

<{Py 1 y

E

E}> we introduce the inner product

Si nee

~ ~

1

L

a: P (x) 1 ::! llP llV(E P) Il

Î

aJ. PyJ. llV(E,P) • j=1 J y j x , j•1

each Cauchy sequence in V(E,P) is pointwise convergent on E. Thus we arrive at a completion of V(E,P) which consists of pointwise limits of Cauchy sequences and which aèlmits P as its reproducing kemel'. D

1. 6. Deftnltion Let K € PT{E).

The space H(E,K) is defined to be the functional Hilbert space induced by K.

We make some remarks.

(1.1) Let K € PT(E). For all x,y € E we have that K(x,y) • K(y,x) ,

2

llKyllH(E,K) • K(y,y) ;:: 0 ,

jK(x,y)i2

~

K(x,x)K(y,y)

(1.2) The pointwise product of two functions of positive type on Eis again a function of positive type. Cf. Lemma 1. 17.

(1.3) Each finite dimensional Hilbert space of functions on a set Eis a functional Hilbert space.

Summarizing: Each function of positive type on E is a reproducing kemel of precisely one functional Hilbert. space on E.

The elements of a functional Hilbert space admit the following simple characterization, which can be found in a monograph of T Ando. Cf.

[And], Chapter II, Theorem 1.1. The characterization is a growth esti-mate condition.

1. 7. Lemma

Let K € PT(E) and let F denote a complex valued function on E.

Then the function F belongs to H(E,K) iff there exists y > 0 such that for all i € lN and o.. € C, y. € E, 1 ~ j ~ i,

J J

Proof:

•. Let i € lN and a. € 4:, y. € E, 1 ~ j ~ i. Then we have

1

j=1

Î

a.. J F (y.) J

12

t 2

l(

I

a..K

,F)

1:;;

j=1 J y j H(E,K)

""· On the linear space V(E,K) we define the linear functional m by

t R,

m(

I

a.. K )

=

I

a. F(y.) • j=1 J yj j=l J J

The assumption implies that m is well defined and that m is bounded

on V(E,K). Therefore m extends to a continuous linear functional on

H(E,K) with Riesz representant G. So, for all y E E

Hence, F =GE H(E,K).

1. 8. Corollary

Let F E H(E,K). Then

2

llFllH(E,K)

= sup { \

Î

a..

"F<Y:T\

2 (

Î

<ik a 3• K(yk,yJ.

>)-1 1

i

Î

aJ.

1),

3• llH(E,K) F

o} .

j=1 J J k,j•1 j=1 Proof:

The statement is equivalent to

llFllH(E,K) = sup { 1(G,F}H(E,K)1 1 G E V(E,K), llGllH(E,K) 1} •

Since V(E,K) is dense in H(E,K), the statement follows.

As already announced in the introduction to this chapter, we introduce a partial ordering in PT(E).

1. 9. Definition Let Kt•Kz € PT(E).

By K₁ ::> Kz we mean K_{2 -K1} E PT(E). ([K

2-K1](x,y)

=

K2(x,y)-K1(x,y).)

a

The next theorem is one of the most useful results in this section. It shows how two functional Hilbert spaces fit in each other.

1.10. Theorem ([Ar], Part 1, 13, _{Corollary IV2)}

Let K₁ and

K:z

belong to PT(E).

a. I f K₁ :;; yK₂ for some y > 0, then H{E,K₁) c;. H{E,K₂) and

Proof:

We put (•,•)i and ll•Ui for (•,•)H(E,K.) and D•DH(E,K.)"

1 l.

a. Let K₁ :;; yK₂• Then for F € H(E,K₁) we have the estimate

Lemma 1.7 implies that F € H(E,K

2) and Corollary

1:a

implies that llFll

2 :;>

./Y

llFll1•

K € V(E,K

1 ). we define

1,yj

itence G • 0 impli~s _{VFEH(E,Ki) : (F ,G ) 2} == 0 and G F 0 implies

G* F

o.

.

For GE V(E,K₁), G

F

O, we define the linear functional !G on H(E,K₁) by !G(F) = (F,G)₁/llG*ll₂, F € H(E,K₁).

For all G E V(E,K₁), G

F

O, we have

HGHI • llGH/RG*llZ , VF€H(E,K

The Banach-Steinhaus theorem implies that there exists

a

> 0 such that VGEV(E,K

1): llGll1 ::; ollG*ll 2, whence for all fl E IN and aj E ~. yj E E, 1 ~ j ~ fl,

Hence K 1

The proof of Theorem 1.10 bas been proved with the aid of Lemma 1.7. The proof in [Ar] is based on different arguments. However, Theorem 1.10 is also an extension of Lemma 1.7.

(1.4) Let F denote a complex valued function on E. The function

~

E

PT(E) is defined by ~(x,y)

=

F(y)F(x), x,y

E

E.

Then H(E,~) is a one-dimensional Hilbert space with orthonormal basis

{F} and

1 _j=1

r

a. _J

F<Nl2

_J

A reformulation of Lemma 1.7 in terms of Theorem 1.10 reads: F E H(E,K) l.• ff 3 _y>O:~•Y· " K

The next lemmas clarify the relations between orthonormal systems in H(E,K) and. the reproducing kernel K.

1.11. Lemma

Let (4>n)nEI denote an orthonormal system in H(E,K) for some K E PT(E). Then

a. For all x,y E E the series lnEI 4>n (y) 4>n (x) is absolutely convergent.

b. The function K41 : Ex E ~ t defined by Kq,(x,y)

=

InEI 4>n(y)4>n~x) belongs to PT(E) and Kq, ~ K.

c. The system (4>n)nEI is an orthonormal basis iff Kq, =K.

1.12. Lemma

Let P denote an ortbogonal projection in H(E,K) and let the function L: E x E -+ ~ be defined by

L(x,y)

=

[P(Ky)](x) , x,y € E • a. Then L belongs to PT(E) and L ~ K.

b. I f P(H(E,K)) is endowed with the Hilbert space structure induced by H(E,K), then P(H(E,K))

=

H(E,L).

Proof:

We only prove part b. For all F € P(H(E,K)) and y € E.we have

Hence, P(H(E,K)) has reproducing kemel L.

The functions of positive type, introduced in the two previous lennnas, are of the same kind.

1 . 1 3. Lemma

Let K,L € PT(E) such that H(E,L) c H(E,K). The following statements are equivalent:

a. There exists an orthonormal system (~n)n€I in H(E,K) such that

V L(x,y)

=

l

~~ (x} •

x,y€E n€I n n

b. VF€H(E,L) : llFllH(E,L) = llFllH(E,K) •

c. There exists an orthogonal projection P in H(E,K) such that

Vx,y€E : L(x,y)

=

[P(Ky)](x) •

Proof:

a. "b. Let G € V(E,L). Then G

=

If.1 aj LYj and