Relations between exponential laws for spaces of C∞ functions

Internat. J. Math. & Math. Sci. VOL. 20 NO. 2 (1997) 209-218

209

RELATIONS

BETWEEN

EXPONENTIAL LAWS

FOR SPACES

OF

Coo-FUNCTIONS

PETERBISTR(M

Department

ofMathematics

/bo

Akademi

SF-20500/bo

Finland

(Received

December6,1992 andin revisedformJune 1,

1995)

ABSTRACT. In

this paperweprove that some new aswellas somealready existing expo-nentiallawsforspaces ofCo- and holomorphic functionscanall begenerated fromonegeneral

exponential law.

KEY WORDS

AND PHRASES.Exponentiallaw, holomorphy,convergencestructure. 1992

AMS

SUBJECT CLASSIFICATION CODE.

58C25.

1.

INTRODUCTION AND PRELIMINARIES.

In

[3]

anexponential law

C(c(U

V,

G)

C(c(U,C((V,

G))

was proved for spaces of Coo-functions

(in

the Bastianisense

[1])

definedonc-open sets inarbitrary convergence vector spaces and taking values in

L-embedded

spaces.

Here,

convergence in

C((V,

G)

means convergence

withrespect to allderivatives,where the linear spacesareequipped with continuousconvergence. The idea inthispaperistousethis exponential law in order to construct exponential laws within other differentiability theories.

We

obtainsuch lawsinthree different theories; asimilartheory as that in

[3]

but with the continuous convergence replacedwith its equable structure, the differential calculus by Seip

[13,14]

for compactly generated spaces and finally the concept of

holomorphyby Bjon

[5,6];

Bjon and LindstrSm

[7].

A

convergence space

X

[8]

is a set, onwhich witheach point x E X is associated aset of

filters,whicharesaidto converge to x, such that thefollowingconditions hold:

1)

Thetrivialultrafilterassociated withx alwaysconverges tox;

2)

If

"

_>

and convergestox, then

"

converges tox;

3)

If

"

and

G

convergetox, then

"

C{ converges tox.

A

convergence vector space

(cvs)

[8]

is aconvergence space withavectorstructure,suchthat the vector operationsare continuous

(a

map is continuous if it preserves

convergence).

Allvector spaces in this paper have the scalar field

]K(=

R

or

C).

A

cvs E issaid tobe equable

[10]

if each filterwhichconvergesto zero inE contains afilter

,

suchthat V{

G

and converges to zero inE.

Here V

denotes the zero-neighbourhood filterof

K.

Clearlythereexists on

E

acoarsestequablevector convergence structure finer than the original structureonE. The vector spaceEendowed with thisequablestructure is denoted

E

.

Alltopological vector spacesareequable.

For

X aconvergence space and

E

a cvs

C(X,

E)

denotes the vector spaceofall continuous

functions

f

X E endowedwith continuousconvergence

[2]. A

filter

Co(X,

E)

iff foreach xEX and each filter

g

which convergestox, the filter

’({}

converges to

zero inE. Wewrite

Ce(X,E)

and

Le(E,F)

insteadof

(Cc(X,E))

and

(Lc(E,F))

.

A

cvs

E

issaid tobe

L.-embedded

if the mapping

JE:

E

L.LE (a

c,

e),

jE(x)l

l(x),

into thesecond dualis anembedding

[4].

If Eis L.-embedded,thesameapplies forthespace

C.(X,E) (a

c,e)

by Bjon

[4].

All Hausdorfflocallyconvex topological vector spaces are

and

L-embedded

and allpolar bornological vector spaces are

L-embedded.

An L.-embedded

cvs isequable. The dual

LE

ofan

L-embedded

cvsE separatesthe pointsof E.

A

convergence space is said to be locally compact ifevery convergent filter contains some

compact set. Let X be a convergence space. The inclusionmappings K

X,

where K

ranges over all compact subsetsof

X,

induce afinal convergence structure on X. The set X

equippedwiththis structureisdenotedX

c.

A

filter

"

converges in

X

"

iff

"

converges inX

and

"

containsacompact set

[2,9].

ThusXtc_{isthecoarsest locally compactconvergence space}

withafinerstructurethanX.

We

canconsider lc asafunctorfromthecategory ofconvergence

spaces to the full subcategory of locally compactconvergence spaces. Thisfunctoriscoreflective,

so

(X

x

Y)=

X

"

x

Y

forconvergence spaces

X

and

Y

[9].

TheopensetsofaconvergencespaceX defineatopology

t(X)

on

X

[8].

This is afunctor between the categories of convergence and topological spaces.

Let

k oIc be the idempotent composedfunctor.

We

callaseparated topological spaceX compactly generated if

X

kX. All

separated locally compact topological spaces and all metrizable spacesarecompactlygenerated

[lS].

If isatopological space,theni"kX

X

iscontinuous, which however not always istrue

when X is aconvergence space.

or

topological spaces X nd

Y

oneusuallywrites X x Y

the

-prou

ofXand instead of

(

x

). Now,

X

x Ydonot in generalequal x

Y,

no

evenifX and

Y

arecompactly generated. Ifhowever

X

iscompactly generated andY isa

locally compacttopological space, thenX

x

Y

X

xY

[15].

A

majoradvantagewhendealingwiththecategory of compactly generatedspaces compared to thecategoryof topological spaces, isthat theformeris Cartesianclosed whereas the latter

isnot.

By

Gabriel- Zisman

(see [13])

wehave

f

C(X, kCo(Y,Z))

.==

]

6_

C(X.xk

Y,

Z),

for compactly generated spaces X, Y and Z. It is mostly because ofthis type ofuniversal

mapping property, that U. Seip has been able to construct his exponential law for spaces of C-funetions in

[13]

and

[14].

2.

SPACES OF DIFFERENTIABLE FUNCTIONS.

Let E

be a cvs.

A

set U

c

E

is saidto be c-open in

E

if it is a unionofcircledopen sets in

E,

orequivalently that to everyxE U thereexists a circledopen set

Uz

such thatx

+

Uz

CU.

Every

open setin atopological vector spaceisc-open.

Let

E and

F

be cvs and U a c-open set in E.

We

say that a function

f

U

F

is

d(erentiable at z U ifthere exists acontinuous linear operator

D](x)

E

F,

such that

themapping

e!

S

U

F

(S

{A

]K:

[A[

<

1}),

definedby

EXPONENTIAL LAWS FOR SPACES OF C -FUNCTIONS 211

is continuous.

As

usual

f

U F is said to be

differentiable

ifit is differentiable at every point inO and continuously

differentiable

orof class

C

ifit isdifferentiableand thederivative

Df

U

La(E,F)

is continuous

(a

c,e).

Higher order derivativesaredefined recursively by using thenotations:

La(E,F)-

F,

La(k+IE,

F)

La(E,

La(kE, F))

fork-0,1,...

We

say that

f

isofclass

C

if

DPf

U

La(PE,

F)

(p

0, 1,

...)

existsandiscontinuous.

For

p

_>

1this is equivalent bysaying that D

f

is ofclass

C

-1.

If

f

has derivativesofall ordersit issaidtobeofclass

C.

It

is astraightforwardexercise toverifythat the functions fog,

f

xg

f

xg(x,

y)

(f(x),g(y))

and

If,

g]

[f,g](x)

(f(x),g(x))

whendefined, areofthesame class

C

as

f

andg are,

likewisethat continuous multilinear mapsare ofclass

C.

In

normed spacesa function is of

class

C

wheneveritisp timescontinuously Fr6chet differentiable.

In

locallyconvextopological vector spaces the class

C

agreeswiththeonedenotedby

C

in

[11].

Let

C(U,

F)

denotethe subspace of

e(u,

F)

consisting of all functions

f

U F of class

C

and

Ca)(U

F)

the set

C(U,

F)

equippedwiththeinitial convergence structure inducedby

the

fferential

operators

D’C:(U,F)--.Ca(U, La(E,F))

k=0,1,...,p,

(k

variesover1N forp

oo).

Thus convergence in

Ca)(U,

F)

meansconvergence withrespect to each existing derivative.

In

order to compare the spaces

C,)(U,F)

and

Cc)(U,F

with each other, we need some

information about the spaces

La(E,

F).

Sinceembeddings turn out to be

useful,

webriefly skiss somegeneralfeatures ofembeddings in function spaces.

If,

fortwocvs Fand

G, F

isembedded inG

(denoted

F

G),

weidentify Fas asubspace

of G. Then also

C(X, F)

Ca(X,

G) (a

c,e)

forevery convergence space

X,

aswell as

L,(E,F)

La(E,G)

(a-

c,e)

foreach cvsE.

By

Bjon

[6]

we

haveV,(X,f’)

C,(X,F)

for

X aconvergence spaceandFa cvs. Obviously

L(E, F’)

L(E,

F)

forcvs EandF.

LEMMA

2.1.

Let

Eand Fbe twocvs. Then thereexistembeddings

L,(E,

F)

(Lc(E,

f)),

L,(E,

f’)

L(E,

F)

fork

_>

1.

Moreover,

if

E

isequable, the embeddings aboveareequalities

L,(E,F)- (Lc(E,F))

,

L(E,F

)

L(E,F)

for k

_>

1.

(2.2)

PROOF.

The embeddings in

(2.1)

canboth be provenbyinduction, the first because of

L.(E,L.(E,F))

L(E,(Lc(E,F))

)

L(E,L(E,F))

and the second similarily. The verysameinduction procedure can be applied to

(2.2),

since

L,(E,

F

)

L(E,F)

forequable E.

We

nowhave the tools neededforproving the following proposition.

PROOF.

Sincethe associatedequablestructure is finer than the originalone,weobviously have

C

(0

r,

F’)

C

C’

(0

r,

F)

C

C

(or,

F).

Then the first embeddingis more orlessadirect

conse-quenceof

Ce(or,

Le(E,

F

))

"--,

C(Or,

L,(E,

F)),

guaranteed by

(2.1).

Thesecond embedding,

onthe otherhand,emergesfrom

C,(U,L,(#’E, F))

C,(U,L(’E,F)),

alsobecauseof

(2.1).

As

shown in

[3],

the mapping

f

(f,

Df,...,Df)

definesanembedding

C,)(U,F)

HCo(U,

Lo(’E,F))

(a =c,e).

k=0

By

Bjon

[4],

theL,-aswellastheL,-embeddedspacesareclosedunderformating ofarbitrary

productsandsubspaces.

Consequently

weobtain:

PROPOSITION

2.3. Let U bea c-opensubset ofacvs E andF an

L-embedded

cvs. Then also

C,)(U,F)

isL,-embedded

(a

c,e)for

p 0,1

3.

EQUABLE

EXPONENTIAL LAW

AND

COMPLETENESS.

Let

G bea cvsandletU beac-open subset ofanequablecvs E.

In

general

C(U,

G

)

isa

proper subsetof

C(U, C).

However,

L(E, C)

L(E,

C).

Sincethe spaces

C(U, G)

somewhat

arebetween

C(U,

G)

and

L(E, F),

itseemsreasonable to question whether

C(U,

G’)

C(U, G)

forsomelargep.

PROPOSITION

3.1. Let G bean

L-embedded

cvsandU ac-open subset ofanequable

cvsE. Then

c,)(u,c’)

c’.)(u,c)

fo,-

,

_2,3 ,oo.

PROOF.

Accordingto Proposition

2.2,

the space

Ce)(U,

G’)

isembedded into

C,)(U,

G).

Thusweonly have toverifythat

C(U, G’)

C(U, G)

forp 2, oo.

Now E

isequable, so

by

(2.2)

there is an equality

C(U,L,(’E,G’))

C(U,L,(’E,G))

fork

>

I. Since

C(U,G)

C

C(U,

G),

it remainsto be shown that

f-

U G"isdifferentiablefor

f

C(U,

G).

This is the

casewhene_I S x

U=

G is continuous for eachx U

(S

{A

]K-IA

<

1}).

Certainly

Sx

U=

is ac-open subsetoftheequablespace

K

xE.

By

proposition2.3in

[6],

e! Sx

U=

G

iscontinuousif

e!

Sx

U=

Gisdifferentiable.

As

intheorem3.4in

[6]

wewrite

el,(,h

"(t,,h)dt,

where

.(t,,h)

DI(

+

th)h- DI()h. Now,

"(t,-,-)

is differentiable with the derivative continuous e function of all three variables

(t,,h).

Thus, by theorem

.

in

[6],

e!

i differentiable ndtheproposition isproved.

PROPONI’rIoN

3.2.

Let

G bean

-embedded

evsandU ae-open subset ofanequable

cvsE. Then

PROOF.

Certainlythestatement is trueforp 0.

We

proceed by inductionand assume that the assertionholdsforp

>_

0.

Let

f

C’+2(U,

G).

We

have toshow that

f

C,’+(U, G).

By

the induction hypothesis

Df

U

L,(’E,

G)

exists and is continuous.

Let

x U. The

function

Df

U

L,(’E, G)

is differentiable atx if

eV!

S x

U=

Le(E, G)

is continuous.

As

in theproofofProposition3.1, wewrite

EXPONENTIAL LAWS FOR SPACES OF C -FUNCTIONS 213

where

f(t,-,-)

w o

[D’+If

o

((t),pr2]

-I-

D’/If(x)

opr:z, andwhere

"Lc(’+IE,

G)

E

L,:(’E,G),

(g,x)=

g(x),

(t):S

U=--*

E,

(t)(s,h)--x

+

tsh,

pr: S x

U=

E,

pr:(s,h)

h,

D+f

U

L(+IE,

G).

Since

Lc(PE,

G)

is L:embedded, we obtain in a similar manner as in Proposition 3.1 that

eo! S U,

L(PE, G)

is differentiable. Thus

Df

U

Le(E,

G)

isdifferentiable and evencontinuously differentiableas aconsequence of

(2.2)

and the hypothesis that

D+f

U

L(+IE, G)

isdifferentiable. Hencetheassertion is trueforpq- and thus for allp 0, 1,

Combiningthe proposition above and Proposition 2.2weobtain:

COROLLARY

3.3.

Let

G bean

L=-embedded

cvs and U a c-open subset ofan equable

cvsE. Then

c,(v,

c)

(co(u,

c))

.

Wearenow ready for the exponential law.

THEOREM

3.4.

Let

G bean

L,-

oran

L:embedded

cvs andU andV c-open subsets of equablecvs

E

and

F

repectively. Then there existsanatural isomorphism

=(u,c,>(v,c)).

c<,)(u

x

v,

G)""

PROOF.

Let

Gbe

L:embedded

and consider its

L:embedded

reflection

G

(i.e.

the canon-icalimage ofG in

L,LcG).

According to

[3]

thereis anexponential law

=(u

v,

co)

"(u,

=(v,

co)),

and thus

(c(o(u

v,

co))"

(C(o%(V,c(v, co)))’.

The setUxVisc-open in theequablecvs

E

xFand

C((V,

Go)

is

Lc-embedded

by Proposition 2.3,sobyCorollary 3.3:

c,(

x

v,

Go)

=-

C,(u, Co(v,

Go)).

(3.3)

Thus,by Proposition 3.1 andCorollary3.3 appliedonthe right member of

(3.3),

weget

(v,

co)).

c,>(u

v,

co)

-

c:>(u,

Since

H

H

for

L:embedded

H,

theassertion follows for the case that the range space is

L:embedded.

UsingProposition3.1 and thefactthat

(G)

G byBjon

[4],

the isomorphism in

(3.4)

finally turns into

c,(u

v,

G)

=-

co(u,

c,(v,

c)).

REMARK. In

generalfor equable convergence spaces

X,Y

andZ

Thereforethe exponential law forequable structures in the theorem above do not fit into the verygeneralsettingof differential calculus, basedoncategory theory, by Nel

[1.].

We

conclude thissectionbygiving somecompletenessresults.

PROPOSITION

3.5.

Let F

be a complete

Lcoembedded

cvs and U ac-open subsetofa

cvs E. Then

Cc)(U,

F)

iscomplete forp 0, 1,

.

PROOF. By

Bjon and Lindstrbm

[7], C(U,

F)

iscomplete ifFis

Lcoembedded

andcomplete. Thus

Co(U,

L(E,F))

is complete for all k

_

0.

Let

p be finite

(the

casewhenp can be treated in the same

manner)

and let

(f,)

be a Cauchy-net in

Cc)(U,F

).

Hence

there are

mappings

f

6

U(U,L(E,F))

(k

0,1

,p)

with

Df,

f

in

C(U,

Lc(E,F))

fork=0,1,...,p.

However,

by theorem3.4in

[6], f

D(f_l)

fork 1 p,by which

Df,

I

Dfo

in

C(U,L(E,F))

fork 0, 1,...,p.

Thusf,

I

f0

in

C’)(U,

F)

and theproofiscomplete.

The completeness of

Ce)(U,

F)

is obviousunder much more restrictive conditions.

PROPOSITION

3.6.

Let F

be acomplete

La-embedded

cvs

(a

c,

e)

and U a c-open subsetofanequablecvsE. Thenthe space

C(e)(U,

F)

iscomplete.

PROOF.

FirstsupposethatFis L,-embeddedand complete and let

(f,)

bea Cauchy-net

in

C()(U,

F).

Althoughwedonotknow whether the spaces

Ce(U,

L(E,

F))

arecomplete, we have the embedding

C,(U,

L,(E,

F))

Ce(U,

L(E,

F)),

and thus

(DL)

is a Cauchy-netin the space

C,(U,

Lc(E,F))

that is complete by Bjon and Lindstrbm

[7].

As

in the proposition aboveweobtainafunction

f0

6

C(U, F)

with

f,

f0

in

(C((U,F))’.

The completeness of

C(e)(U,F)

then follows from Corollary 3.3.

Now

assume that

F

is

L,-embedded andcomplete. Then

F

isclosed in

L,LeF

andconsequently

C(e)(U,

F)

canbe con-sideredasa closedsubspace of

C(,)(U,

LeLeF).

Let

i"

LeL,F

C(e)(LeF,

K)

be the inclusion.

Its

associated mapping

"

LeL,F

x

LeF

K

isbilinearandcontinuous, hence ofclass

C.

By

Theorem 3.4also is ofclass

C,

and espe-ciallyitis continuous. Since

LeLeF

isclosedin

Ce(L,F),

wethus obtain that

LeLeF

is closed

in

C(e)(LeF,

IK)

too. Consequently

C(e)(U,L,LeF)

is aclosed subspace of

However,

since

LF

isequable,

C(,)(U,

C(e)(L,F,

IK))

isisomorphic to

e(e(U

xLeE,

K)

by Theo-rem3.4.

But

thislast spaceiscompleteaccording to the firstpart ofthisproof.

Because

closed sets incompletespaces arecomplete,we havethusshown the completenessofthecvs

C(,)(U,

F).

4.

COMPACTLY

GENERATED

SPACES.

A

compactly generatedvector space

E

is saidtobeak-vectorspaceifthevector operations

EXPONENTIAL LAWS FOR SPACES OF C -FUNCTIONS 215

is a k-vectorspace.

A

k-vectorspaceisnotalwaysatopological vector space,sincethestructure of the k-productingeneralisfiner than thatof the topologicalone. If, however, Eisak-vector

space, thenEc is a cvs.

To every k-vector space

E

can beassociated a locally convex topological vector spacecE,

where thetopologyisgenerated by the convex zero neighbourhoods inE

[13]. A

k-vector space

Ewith

E

kcE,wherecEissequentiallycomplete,wecalla convenientk-vectorspace. These convenient k-vectorspaces

(containing

all Frchet

spaces)

constitutethebasisfor the differential calculusby Seip

[13,14].

Let E

and

F

beconvenientk-vectorspaces andUanopen subsetofE.

A

function

f

U

F

is said to beofclass

C

if thereis a continuousmapping

Df

U

kLco(E,

F),

such that the map go" 0 F

(0

{ (s,x,h

E

IK

x

U

x

E"x

+

sh

U}),

definedby

o(O,,)

o,

o(,,,)=

D

f(x)h

fors:O,

iscontinuous. Furtherwe say that

f

U F isof class

C’

ifD

f

U k

Lo(E,

F)

isof class

C

-1.

By

C(U, F)

wemeanthe space of all

C-functions

f

U F. Weusethe following notations

kLo(E,F)

F,

kLo(n+IE,

F)

kLo(E, kLo(nE,

F))

forp 0,1

Theset

C(U, F),

equippedwiththeinitialstructureinducedby the differential operators

D"

C(U, F)

kCco(U,

kLo(E, F)),

p

O,

1,2,

wedenoteby

init(C(U,F)).

Let

C(%(U,F)

k(init(Co(U,F))).

In

[13]

and

[141

it isshown that

C((U,

F)

is a convenient k-vectorspace andfurtherthatthere exists anexponential law

c<%(v

v,

e)

-

C%(u, c(v,

G))

(4.1)

for convenient k-vectorspaces

E,

F andG withopen setsU and V in

E

andF respectively.

In

order to compare the classesof

C-

and

Cc-functions,

weneed thefollowing lemma.

LEMMA

4.1.

Let

X becompactlygenerated andY atopological space. Then

c(x,Y)

c(x,Y)

c(x’,Y)

c(x’,Y),

(4.1)

Coo(X,

Y)

Coo(X,

z).

Moreover,

ifY is aseparated topological vectorspace,then

Co(X,

Y)

C,(X",

Y).

(4.2)

(4.3)

PROOF.

Since the verification of

(4.1)

is straightforward and thus left to the reader and

(4.2)

is a theorem due to Steenrod

[15],

weonlyhave to concern about thestructure in

(4.3).

Let

"

0 in

Co(X, Y),

x

.

X

and{beafilter with

G

x in

X

c.

Thenthere is acompact set

K

G.

Hence,

to everyzero neighbourhood U ofY thereexists anF

"

with

F(K)

C U

and thus

’()

I

0 in

Cc(X

=,

Y).

On

thecontrary, let

1

0 in

C(X

c,

Y),

Kbeacompact set inX andU asabove.

Let

furtherx K and

g

beafilterwith

g

I

x in

X;’.

Thus thereexist

many sets

G=,

thatisK C

G= t2...tJG=.

Let

F

F=I

f...

NF=.

ThenFE

"

and

F(K)

C U,

so

"

0in

Co(X, Y).

In

comparing

C-

and

C-differentiability,

the equality

(4.3)

willbeof fundamental impor-tance. The

C-functions

are defined onopen sets whereas the

C-functions

require c-open domains.

Now,

itis easy to see that everyopen set ina k-vectorspace alsois c-open. Since X

andX

=

has exactly the same opensets,flt=

isc-open inE

=

if Uisanopen subset ofak-vector

space E.

PROPOSITION

4.2.

Let

U beanopen subset ofa k-vectorspace

E

and F aseparated locally convex topological vector space. Then

C"(U,F)

C(U=,F)

C(U, kF).

(4.4)

PROOF.

By

repeateduseof

Lemma

4.1 wearrive in

kLo(’E,

F)

kL=(’E

,

F)

kL,o(’E, kF),

forp 1, 2

By

definition,

f

.

C(U,

F)

iffforeachp 0, 1, there existcontinuous mappings

D’+lf

U

kLo(’IE,

F)

and

g:

5

kLo(’E,F)

(5

{(s,x,h)

.

]K V

x

E x

+

sh

e

V}),

where thelatterisdefinedby

D’

f (x

+

sh)

DV

f (x)

D+:f(x)h

fors 0.

Takeafunction

f

C

(U, F).

Since

C(U,

kLo(’E,F))

also

gp

()"

L(’E

,

F)

is continuousforeveryp,andhence obviously

f

.

C(U’,

F).

It

thenremains tobe shown that

C(U

,

F)

C

C(U,F).

Since the procedure isthe samefor

.lrr

F).

As

in allderivatives,itsuffices to be verified that

C(U

F)

C

C(U,F)

Let

f e

theorem 3.4 in

[6]

wewrite

Since

F

is L-embedded,by lemma .2 in

[6]

g"

(}

N is continuous, where is considered to bea subspace of

K

U x E

=

(K

x U x

E)

,

and thus also when

d

is asubspaceof

IK

x U x

E.

Hence

J"

(U, F),

whichcompletes theproof.

PROPOSITION

4.3.

Let

U be anopen subsetofak-vector space

E

and

F

a separated locallyconvextopological vector space. Then

’

(u,

F).

kc(o)(u

F)

PROOF.

Since

kCo(U, kLo(’E, F))

kCo(U, kL=o(’E, kF)),

(4.6)

follows directly from

(4.4).

Further

(4.7)

isaconsequence of

(4.4)

and the relation

EXPONENTIAL LAWS FOR SPACES OF C -FUNCTIONS 217

THEOREM

4.4.

Let

E, FandG beconvenient k-vector spacesandU aswell asVopen subsetsofE andF respectively. Then thereisanexponential law

c,%(u

v,c)

_

c>(u, c(,%(v,c)).

(4.8)

PROOF.

SinceG kcG, where cG is aseparated locally convextopological space, cG is

L=-embedded

andhence,by theorem 3.9in

[3],

thereisanexponential law

(4.9)

The law in

(4.8)

thenfollowsfrom that in

(4.9)

by repeateduseof

(4.6)

and

(4.7),

remembering that

C(=)

(V

,

cG)

istopological and

L:embedded

and thusaseparatedlocallyconvextopological vector space.

5.

HOLOMORPHY.

anopen subset ofE.

A

function

f

U Fissaidtobe

Gateaunc

holomorphic if the function

A -+

l(f(z

+

Ah))

is holomorphic inaneighbourhood ofzeroin

C

foreachx (

U,

h ( E and

ELF. It

is holomorphic, ifit is

Gateaux

holomorphic and continuous.

Let

H(U, F)

denote the setofall holomorphic functions

f

U F.

Let

U be a --open subset

(i.e.

to each x E U there is a circled convex open set

U=

with x

+

U=

C

U)

ofacvs

E

and

F

an

L:embedded

sequentiallycompletecvs. Bjon and LindstrSm

[7]

show thataholomorphic function

f"

U Fhasanexpansion

f(x

-i-

h)---

.

m!

h,

(5.1)

’t----’O

where the mappings

"’f(z)

E

F,

definedby

m!

/I),

f(x

+

Ah)dA

(m

0,1

),

(5.2)

jf(=)=

=

+

arecontinuousm-homogeneous polynomials.

Certainlyadifferentiable function is holomorphic.

On

the contrary,by corollary 4.1.1 in

[6],

aholomorphic function

f

U

F,

where U is a--open subset ofa cvs E

(respectively

an

equable cvs

E)

and

F

is asequentially complete

L:embedded

(Le-embedded)

cvs, is of class

C

(C)

withthe derivative givenby

(5.2)

form 1;that is,

1

f

/(x-I- Ah)

D

f(x)h

JP,

lffi,

A

dA.

(5.3)

Thus

C:(U,

F)

H

(U, F)

(a

c,

e),

p 1,2, oo.

(5.4)

Let

(f,)

beanetwithf,

I

0 in

H=(U, F).

Byproposition4.5 in

[7],

the mapping

J"

H,(U, F)

U x E F,where

j(f,x,h)

Jf(z)h,

is continuous. Thus

Dr,

0in

C,(U, Lc(E,F)).

From

(5.4)

we get that

Dr,

H(U,L(E,F))

and hence

D2f,

], 0 in

Cc(U,

Lc(2E,

F))

etc. If

E

is equable and

F

is

Le-embedded

and sequentiallycomplete,wecan usetheorem4.3in

[6]

andget

PROPOSITION

5.1.

Let

U be a z-open subset ofacvs

E

(an

equable cvs

E)

and

F

an

L

c-embedded (L,-embedded)

sequentiallycompletecvsF. Then

C)(U,F)

Hc(U,F),

(C,)(U,F)

H,(U,F))

p= 1,2,...,oo.

As

iseasilyseen, completenesscanbereplacedwith sequentiallycompletenessinPropositions 3.5 and 3.6. Thus the theorem below follows directly from Proposition 5.1.

THEOREM

5.2.

Let

Gbeasequentiallycomplete L,-embeddedcvs

(a

c,

e)

andUand

Vr-open subsetsoftwocvsE andFrespectively, which areequable whena e. Then

,,(u

x

v,c)

H,,(U, Ho(V,C))

(,,

,,).

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1.

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Applications

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2.

BINZ,

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Continuous

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3.

BISTR6M,

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Kobe

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BJON, S.,

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BJON, S.

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Infinite

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