Ideals of Holomorphic Functions on Fréchet Spaces

Advances i n HoZomorphy, J.A. Barroso led. )

0 North-Holland Publishing Company, 1979

IDEALS OF HOLOMORPHIC FUNCTIONS ON F ~ C H E TSPACES JORGE MUJICA

1. INTRODUCTION

In this paper we are concerned with the study of certain ideals of holomorphic functions or holomorphic germs on infinite dimensional spaces.

More precisely, we consider

the ideal generated by a family of holomorphic functions without common zeros.

And likewise for holomorphic germs.

Throughout the paper we keep in mind the following finite

Let

dimensional result due to Cartan [l]:

#(U)

denote the

algebra of all holomorphic functions on a domain of holomorphy U

in - Cn,

and let

without common zeros. dense in

#(U)

the family

3

a

C

#(U)

be a family of functions

Then the ideal

8

generated by

$

for the topology of compact convergence. is finite, then the ideal

8

equals

is -

If

#(U).

Sections 2 and 3 of this paper are of preparotory nature.

In section 2 we establish some properties of certain

classes of topological algebras,

In section 3 we collect

some results concerning approximation of holomorphic functicns by continuous polynomials. The main results appear in sections 4 and 5 .

563

In

5 64

J. MUJICA

4

section

we d e a l w i t h t h e a l g e b r a

germs on a p o l y n o m i a l l y

convex compact s u b s e t of a Fre'chet

space with the approximation property. e v e r y complex homomorphism on p o i n t of

K.

of a l l holomorphic

#(K)

Then we prove t h a t

i s an e v a l u a t i o n a t a

#(K)

A s a c o r o l l a r y we g e t t h a t t h e i d e a l g e n e r a t e d

by a f a m i l y of germs i n

w i t h o u t common z e r o s , e q u a l s

#(K),

n(K). I n s e c t i o n 5 we d e a l w i t h t h e a l g e b r a

of a l l

#(U)

holomorphic f u n c t i o n s on a p o l y n o m i a l l y convex domain i n a Fre'chet s p a c e w i t h t h e a p p r o x i m a t i o n p r o p e r t y . d e n o t e s t h e Nachbin t o p o l o g y on every

T

w

a

w i t h o u t common z e r o s , i s dense i n the ideal family

8

a

# #(u)

8

u)

#(U)

i s an

A s a c o r o l l a r y we g e t t h a t t h e

U.

g e n e r a t e d by a f a m i l y

d

r

t h e n we prove t h a t

- c o n t i n u o u s complex homomorphism on

e v a l u a t i o n a t a p o i n t of ideal

#(U),

If

need n o t e q u a l

of f u n c t i o n s i n #(U)

#(U)*

for

- u )

T

.

#(U), I n general,

I n t h e , c a s e where t h e

i s f i n i t e , we do n o t know whether t h e s i t u a t i o n can a c t u a l l y o c c u r .

F o r t h e convenience of t h e r e a d e r we have made t h i s p r o v i n g most o f t h e r e s u l t s we

paper n e a r l y s e l f - c o n t a i n e d , use.

The o n l y e x c e p t i o n s a r e theorem 3.2,

S c h o t t e n l o h e r i n [ll], and theorem 5 . 1 ,

2.

proved by

proved by us i n [ y ] ,

LOCALLY m-CONVEX ALGEBRAS AND Q-ALGEBRAS By an a l g e b r a we mean t h r o u g h o u t t h i s p a p e r a

commutative complex algqbrar w i t h u n i t e l e m e n t . homomorphism on a n a l g e b r a

A

By a complex

we mean a n a l g e b r a homomorphism

IDEALS

T: A

+

OF HOLOMORPHIC

565

ON FRECHET SPACES

FUNCTIONS

which i s n o t i d e n t i c a l l y z e r o .

C

A t o p o l o g i c a l a l g e b r a i s a n a l g e b r a and a t o p o l o g i c a l

v e c t o r space s u c h t h a t r i n g m u l t i p l i c a t i o n i s s e p a r a t e l y continuous.

The spectrum o f a t o p o l o g i c a l a l g e b r a

s e t o f a l l c o n t i n u o u s complex homomorphisms on

A

is the

A.

I n t h i s s e c t i o n we e s t a b l i s h some p r o p e r t i e s of two i m p o r t a n t c l a s s e s of t o p o l o g i c a l a l g e b r a s , t h e l o c a l l y m-comx a l g e b r a s and t h e Q - a l g e b r a s ,

t o be d e f i n e d below.

The main

r e f e r e n c e s f o r t h i s s e c t i o n a r e [ 5 ] and 1 2 3 . A t o p o l o g i c a l a l g e b r a i s l o c a l l y m-convex

2 . 1 DEFINITION

i t h a s a 0-neighborhood b a s e c o n s i s t i n g of convex s e t s with

V

2

if

V

c V.

Thus t h e t o p o l o g y of a l o c a l l y m-convex a l g e b r a g e n e r a t e d by a d i r e c t e d f a m i l y of seminorms p(xy) g p ( x ) p ( y )

for all

x, y

in

p

A

is

such t h a t

P r o p o s i t i o n 2.2

A.

below i s proved i n [ 2 , p.

163, P r o p . l . 4 1 .

2 . 2 PROPOSITION

be a l o c a l l y m-convex a l g e b r a . Then:

(a)

Let

A

E v e r y c l o s e d i d e a l of

i s contained i n a c l o s e d

A

maximal i d e a l . (b)

Every c l o s e d maximal i d e a l of

i s t h e k e r n e l of a

A

c o n t i n u o u s complex homomorphism, PROOF

The t o p o l o g y of

of seminorms in

A.

and l e t

p

Let A

P

A

such t h a t

(A,p)

i s g e n e r a t e d by a d i r e c t e d f a m i l y p(xy)

S

p(x)p(y)

denote the alge b ra

A,

for a l l

x, y

seminormed by

d e n o t e t h e c o m p l e t i o n of t h e normed a l g e b r a

(A,p)/p”(O),

Thus i f

mapping, t h e n

n (A) P

rr : A P

3

A

P

denotes the canonical

i s a d e n s e s u b a l g e b r a o f the Banach

p,

J. MUJICA

566 algebra

and

6

A

P‘

Let

I

>

such t h a t t h e open s e t

0

be a c l o s e d i d e a l of

I.

d i s j o i n t from

Then

r(

P

np(I)

of

A

P

.

Since

A

A

nP(I)

of

I

i n

i s contained i n t h e k e r n e l of

c o n t i n u o u s complex homomorphism o n I

i s maximal,

2.3

then necessarily

A

i s a closed i d e a l

P

np(I)

i s contained

A.

Toll

P’

T

on

A

T h i s proves ( a ) .

I = k e r Ton

P’

P

.

which i s a If

proving (b).

A t o p o l o g i c a l a l g e b r a i s a Q-algebra

DEFINITION

is

and i s

r(,(A)

i n t h e k e r n e l o f a c o n t i n u o u s complex homomorphism Then

p

o f c e n t e r 1 a n d r a d i u s S.

P

i s a Banach a l g e b r a ,

P

p(x-1) < g)

Ex E A :

( I ) i s a n i d e a l of

d i s j o i n t from t h e open b a l l i n Thus t h e c l o s u r e

Then t h e r e e x i s t

A.

if

the

i n v e r t i b l e e l e m e n t s form a n open s e t .

2.4 PROPOSITION

hood

If V

the closure of every ideal

I n p a r t i c u l a r , e v e r y maximal i d e a l i s c l o s e d .

is an ideal.

PROOF

I n a Q-algebra,

A

i s a &-algebra, t h e n t h e r e e x i s t s a n e i g h b o r -

of 1 c o n s i s t i n g o n l y of

i s a n y i d e a l of

A,

then

c o n s e q u e n t l y the c l o s u r e

2 . 5 COROLLARY

I

I

i n v e r t i b l e elements.

i s d i s j o i n t from

of

I

V,

If

I

and

c a n n o t c o n t a i n 1.

If a t o p o l o g i c a l a l g e b r a i s b o t h l o c a l l y m-

convex a n d a Q - a l g e b r a , t h e n e v e r y complex homomorphism i s continuous. P r o p o s i t i o n 2 . 6 b e l o w gives a l a r g e c l a s s o f t o p o l o g i c a l a l g e b r a s which a r e b o t h l o c a l l y m-convex a n d Q-algebras. P a r t ( b ) of p r o p o s i t i o n 2 . 6 i s proved i n [ 2 , p.165, A s f o r p a r t ( a ) , t h e proof

of [ 7 , Th.7.11;

Prop.151.

i s a n a b s t r a c t v e r s i o n of the proof

s e e a l s o [ 8 , Th.11.

567

I D E A L S OF HOLOMORPHIC FUNCTIONS ON FRIfCHET S P A C E S

2 . 6 PROPOSITION

Let

be a n a l g e b r a which i s t h e l o c a l l y

A

convex i n d u c t i v e l i m i t of a n i n c r e a s i n g s e q u e n c e o f normed

u A j=1 j ' m

algebras

A

with

j

A =

i n c l u s i o n mapping

A . CA j+l J

and suppose t h a t e a c h

h a s norm one.

Then:

i s a l o c a l l y m-convex a l g e b r a .

(a)

A

(b)

If each

A

i s a Banach a l g e b r a , t h e n

j

is also a

A

Q-algebra.

( a ) Let

PROOF 0

<

6

j

of center

0

c .

be a s e q u e n c e of numbers

j =O

j

let

U

and r a d i u s

Sj,

and l e t

For each

1.

I:

m

(cj)

with

J

d e n o t e t h e open b a l l i n A

j

d e n o t e t h e convex

U

m

u

h u l l of

j=1

ponding s e t s

Uj. U

A s t h e sequence

form a 0-neighborhood b a s e i n

i s c e r t a i n l y convex.

U

We c l a i m t h a t

U

2

Each

A. Given

C U..

U

x, y

in

we can w r i t e

.

where t h e sums a r e f i n i t e and

C Clk = 1, k

xj

E

Uj,

E

~k

and s i n c e

E

Fix

Since

U

J

Uk*

XY =

xy

varies, the corres-

(E j )

c

j ,k

i s convex and

j, k

A

j

and assume

j

s k.

uk

2

0,

xj

Clk

C

hlrk

= 1,

x j Yk

= 1,

xjyk

Then

C k j

Thus

j ,k i t s u f f i c e s t o show t h a t

U

z 0,

E U

xjyk

E

t o show t h a t for all

Ak

i s a Banach a l g e b r a , e v e r y e l e m e n t o f

j , k.

and

1

+

V

j

is

J. MUJICA

5 68

Let V = LJ Vj. Since j' j=1 is a convex 0-neighborhood in

invertible in every

V

j,

1+V

element of Let

W

in

A

A

x

is invertible in

E A

be invertible.

such that

= x( l+x-'h)

E V

x'lh

V. c J

A

v ~ for + ~

and every

A. We choose a 0-neighborhood

for every

is invertible for every

h

h

E W.

Then

x+h =

E W.

3 . POLYNOMIAL APPROXIMATION ON COMPACT SETS

E

From now on

stands for a complex Fre'chet space U

with the approximation property, sets of

K

E, and

U,

functions on

V

denote open sub-

denotes a compact subset of

E.

denote the algebra o f all holomorphic

u(U)

Let

and

and let

continuous polynomials on

P(E) E.

denote the subalgebra of all

We refer to the monographs by

Nachbin [9] and Noverraz [lo] for the definitions and basic properties of holomorphic functions and polynomials on infinite dimensional spaces.

3.1 DEFINITION

For

K C E

we define

We say that K is polynomially IIPIIK = sup lP(x) 1 . x€K convex if K = 2. We say that U C E is polynomially convex

where

if

i n u

is compact for every

'3.2 THEOREM

function K

f

Let

K C E

K

c

be polynom ally convex.

Then every

which is holomorphic on an open neighborhood of

can be approximated uniformly on

nomials on

U.

E.

K

by continuous poly-

569

IDEALS OF HOLOMORPHIC FUNCTIONS ON F a C H E T SPACES

Theorem 3.2 is due to Schottenloher; for the proof see

[ 11, Th.21.

A weaker form of this theorem had been previous-

ly given by Noverraz; see [lo, p.76, Th. 4.3.21.

Theorem 3.2

yields the following corollary; see [ll, Cor.31.

3.3 COROLLARY

ii c u

for every

PROOF U

K c

Let

U C E

Let

be polynomially convex.

K c U.

u,.

Since

ii

U (k\U),

2n

U

and

U

is also compact.

f , holomorphic on a neighborhood of

by putting f

f

equal to one on a neighborhood of

every

IIf-PIl; < 1/2. x E k\U.

We define

r?

(Gnu) U

=

equal to zero on a neighborhood of

Theorem 3.2 there exists a continuous polynomial that

2\~.Since

is compact, s o is

n

is polynomially convex

a function

Then

This yields

r?\U.

P

such

llPllK C 1/2 C IP(x)l

This is a contradiction, unless

By

for

t\U

is

empty. We also need the following standard lemma.

3.4

LEMMA

each

f: E

Let

K c E

and

E

+

>

C

u: E

+ E

such that

Since

K

is compact,

f

<

If(x)-f(y)l

E

g

for every

x E K

and

x.

in

E

with

such that x-y

W.

has the approximation pro erty, there exists a

x-u(x) E W

E

W

y E E

continuous linear mapping of finite rank

x

C g

is uniformly continuous on

and thus there exists a 0-neighborhood

Since

(f(x)-fou(x)l

x E K.

for every

K

Then for

there exists a continuous linear

0,

mapping of finite rank

PROOF

be a continuous function.

for all

x E K.

Thus

u: E

f(x)-fou(x)l

+ E < e

such that for all

J. M U J I C A

57 0 Pf(E)

Let

f i n i t e t y p e on

i . e . t h e s u b a l g e b r a of

E,

E'

by t h e d u a l

d e n o t e t h e a l g e b r a of a l l p o l y n o m i a l s of

E.

of

P(E)

generated

3.4 y i e l d s t h e f o l l o w i n g

Then lemma

corollary.

Pf(E)

3 . 5 COROLLARY

P(E)

i s dense i n

f o r t h e t o p o l o g y of

compact c o n v e r g e n c e , PROOF

P E P(E),

Let

B y lemma

K C E

and l e t

u: E + E

such t h a t

F = u(E)

<

C

for all

Since

O i E F'

such t h a t P(Y) =

y E F.

n

c

C$qY)I

x C E

mi

i=1

Hence

i= 1 for all

be g i v e n .

0

E Pf(E).

Pou

i s f i n i t e dimensional, t h e r e e x i s t

(i=l,.,,,n)

for a l l

>

IP(x)-Pou(x)l

Thus i t s u f f i c e s t o p r o v e t h a t

x E K.

8

3 . 4 t h e r e e x i s t s a c o n t i n u o u s l i n e a r mapping o f

f i n i t e rank

4,

and

and

1.

Pou E P f ( E ) .

I D E A L S O F HOLOMORPHIC GERMS

For

V C E

we l e t

d e n o t e t h e Banach a l g e b r a

#"(V)

o f a l l bounded h o l o m o r p h i c f u n c t i o n s o n t h e supremum.

Then for

of h o l o m o r p h i c germs on

K

c

K,

we define

w i t h t h e norm of

#(K),

t h e space

a s t h e l o c a l l y convex i n d u c t i v e

l i m i t of the Banach a l g e b r a s t h e open n e i g h b o r h o o d s of

E

V,

K.

#"(V)

,

Since

where

E

V

v a r i e s among

i s m e t r i z a b l e , we

571

IDEALS OF HOLOMORPHIC FUNCTIONS ON FRE?CHET SPACES

K.

d e c r e a s i n g f u n d a m e n t a l s e q u e n c e o f open n e i g h b o r h o o d s o f Then p r o p o s i t i o n 2 . 6 y i e l d s t h e f o l l o w i n g ,

4 . 1 PROPOSITION

g(K)

K c E,

F o r any

i s a l o c a l l y m-convex

a l g e b r a and a Q-algebra.

We have a l r e a d y g i v e n a d i r e c t proof t h a t

4.2 REMARK

i s l o c a l l y m-convex;

[ 7 , Th.7.11

see

o r [ 8 , Th.11.

#(K)

On t h e

o t h e r hand, i t i s q u i t e easy t o give a d i r e c t proof t h a t # ( K )

E

i s invertible,

i s a Q - a l g e b r a , as f o l l o w s ,

If

f

then

K,

and t h e r e f o r e t h e r e e x i s t s

E

>

d o e s n o t v a n i s h on

f

such t h a t

0

b = { g

E

> c

If(x)l

# ( K ) : )Ig-fllK

<

#(K)

for all

Then t h e s e t

x E K.

i s a neighborhood of

C)

i n g o n l y o f germs which do n o t v a n i s h on

K.

f

Thus

consistconsists

b

o n l y of i n v e r t i b l e germs. Next w e c h a r a c t e r i z e t h e s p e c t r u m o f p o l y n o m i a l l y convex.

#(K)

I n t h e p r o o f of t h e o r e m

when

K is

4.3 b e l o w a n d

i t s c o r o l l a r y 4.4 w e u s e most o f t h e t o o l s d e v e l o p e d i n s e c t i o n s 2 and 3.

4 . 3 THEOREM

Let

K C E

a complex homomorphism point

E

a

PROOF

K

for all

on

t h e r e e x i s t s a unique

#(K),

T(f) = f(a)

f o r every

f

E #(K).

14,

4 . 1 and c o r o l l a r y 2 . 5 ,

T

continuous.

>

T

T h e n , given

The p r o o f i s b a s e d on i d e a s of I s i d r o ; d e e

a n d Prop.41.

c(V)

such that

b e p o l y n o m i a l l y convex.

0

By p r o p o s i t i o n Hence f o r e a c h

V 3 K

Prop.3

is

t h e r e exi.sts a constant

such t h a t

f E #"(V).

and l e t t i n g

n +

LO

IT(f)l 5 c(v>

IIfllV

Replacing

by

we get t h a t

f

fn,

taking nth root

J. MUJICA

57 2

for all

f

#"(V).

And since

V 3 K

is arbitrary this

implies that

< lIfllK for all

f

E #(K).

restriction of

Thus we get in particular that the

T

compact convergence. p.205,Th.l],

for all

E'

to

is continuous for the topology of

Hence, by the Mackey-Arens Theorem [: 3 , a E E

there exists a unique point

9 E E' ,

generated by

E'

Since

,

Pf(E)

such that

#(K)

is the subalgebra of

it follows that T ( P ) = P(a)

for all dense in

P E Pf(E). P(E)

But since by corollary 3 . 5 ,

P,(E)

is

for the topology of compact convergence, we

get that T(P) = P(a) for all

P E P(E).

But this implies that

*

/)'(TI

= for all

P E P(E)

and therefore

Schottenloher theorem 3 . 2 , each

K

ed uniformly on

a E

lIP/IK

= K.

f E #(K)

Finally, by

can be approximat-

by continuous polynomials on

E.

Thus we

get that

T(f) = f(a) for all

f E #(K).

4.4 COROLLARY

Let

K

C E

be polynomially convex.

Let

573

I D E A L S OF HOLOMORPHIC FUNCTIONS ON FFfE!CHET SPACES

$

be a f a m i l y of germs w i t h o u t common z e r o s i n

c H(K)

Then t h e i d e a l

Thus t h e r e a r e germs such t h a t

Since

f E 3,

ness o f

and

3

equals

g l , . ..,g

#(K).

E #(K) K.

i s c o n t a i n e d i n a maximal i d e a l

8

h

i s c l o s e d , a n d i n view of

4.3,

is the evaluation a t

T

f E h,

Therefore, each

would v a n i s h a t t h e p o i n t

4 . 5 REMARK corollary

then

B y theorem

T.

#(K)

i s t h e k e r n e l of a c o n t i n u o u s complex

h

a E Kr

some p o i n t

fp E

i s a Q-algebra,

#(K)

homomorphism

...,

in

P f p g p = 1 on a neighborhood of

+...+

proposition 2 . 2

each

fl,

8 f #(K)

If

PROOF

h.

flgl

3

g e n e r a t e d by

8

K.

a,

and i n p a r t i c u l a r a contradiction.

It makes no d i f f e r e n c e ' t y h e t h e r t h e f a m i l y

in

3

4.4 i s f i n i t e o r i n f i n i t e , f o r due t o t h e compact-

K,

if

common z e r o s i n

3

i s a n a r b i t r a r y f a m i l y o f germs w i t h no

K,

t h e n t h e r e always e x i s t s a f i n i t e sub-

f a m i l y w i t h t h e same p r o p e r t y .

5 . IDEALS OF IIOLOMORPHIC FUNCTIONS

Let

U C E.

compact s e t

K C U

e x i s t s a constant

A seminorm

i f f o r each c(V)

> 0

p

on V

#(U)

with

such t h a t

The Nachbin t o p o l o g y on

i s p o r t e d by a

K C V C U, p ( f ) 5 c(V) #(U),

there

)IfllV

for

d e n o t e d by

all

f E #(U).

Tu,

i s t h e l o c a l l y convex t o p o l o g y d e f i n e d by a l l s u c h s e m i -

norms.

5-11THEOmM vex a l g e b r a .

F o r any

U C E,

( # ( U ) ,Tu)

i s a l o c a l l y m-con-

J. M U J I C A

574

F o r a p r o o f w e r e f e r t o [ 7 , Th.7.21 a l s o t h e announcement i n [ 6 ] . t h e s p e c t r u m of

Theorem 5 . 2 b e l o w c h a r a c t e r i z e s

when

(#(U),T~)

see

o r [ 8 , Th.21;

u

i s p o l y n o m i a l l y convex.

4.3 and w i l l

The p r o o f i s s i m i l a r t o t h a t o f t h e o r e m

be o m i t -

ted.

Let

5.2 THEOREM

U C E

be p o l y n o m i a l l y convex.

a c o n t i n u o u s complex homomorphism e x i s t s a unique p o i n t f

a

E

T

on

there

(#(U),T~),

such t h a t

U

Then, given

= f(a)

T(f)

for all

E #(U).

5.3 COROLLARY

Let

common z e r o s i n

The p r o o f

PROOF

(#(U),Tu))

g e n e r a t e d by

8

i s s l i g h t l y d i f f e r e n t from t h a t o f c o r o l l a r y

i s not a Q-algebra.

If

w e r e a proper i d e a l ,

then

c o n t a i n e d i n a c l o s e d maximal i d e a l o f p r o p o s i t i o n 2.2.

Thus

! !

f

5.2,

a contradiction.

&!

(#(U),T~).

a E U,

i s not t r u e ,

not even i n f i n i t e dimensional spaces. i n Corollary

gp(U

g cop

E #(C)

= l7- (1 n= p

x

by

-7 ) n

(A E c).

u)

I n other

T h i s example i s a n

a d a p t a t i o n o f a n example o f C a r t a n ; see [ l , p . 6 0 ] . define

f o r (#(u),T

5.3 i s n o t n e c e s s a r i l y

c l o s e d , a s t h e f o l l o w i n g example shows.

p = 1,2,3,...

Hence

by theorem

The e x a c t a n a l o g u e o f c o r o l l a r y 4 . 4

8

of

would b e

5.4 REMARK

words, t h e i d e a l

8

would be c o n t a i n e d i n t h e k e r n e l

would v a n i s h a t a f i x e d p o i n t

all

the closure

( # ( U ) , T ~ ) , by

of a c o n t i n u o u s complex homomorphism on

E

in

3

(#(U),Tu)).

4.4 f o r ( # ( U ) , T ~ ) in

Then t h e i d e a l

U.

i s dense i n

#(U)

be a f a m i l y of f u n c t i o n s w i t h o u t

U C E

For

)

575

IDEALS OF HOLOMORPHIC FUNCTIONS ON F ~ C H E TSPACES

0 E E',

Let

#

@

and define

0,

Thus a0

fp(X) = Each E C

x E E

2 x = n xo

+

y

0. Thus

x = Axo

f

P

fp = gpo@.

by

(x E E).

+

y,

where

= l,X

@(xo)

fp vanishes only at the points

n = p, p+l, p+2,

with

Thus the functions in

n=p

(1

can be written

y E Ker

and

iT

fp E #(E)

(p=1,2,3,.. . )

...

and

y

E Ker

@.

have no common zeros

E, but the functions in any finite subfamily have in-

finitely many common zeros.

Hence the constant function 1

does not belong to the ideal generated by the functions In the example above, the family finite.

is in-

Thus the following problem remains open.

5.5 PROBLEM family

m

f P'

3

Under the hypotheses of corollary 5 . 3 ,

is finite, does it follow that the ideal

generated by

3

equals

if the 8

#(U)?

REFEm NCE S

1. CARTAN, H. Id6aux et modules de functions analytiques de

variables complexes, Bull, SOC. Math. France 78

(1950),

29-64. 2. GUICHARDET,A.

Special topics in topological algebras,

Notes on Mathematics and its Applications, Gordon and Breach, New Y o r k , 1968.

3. HORVATH, J.

Topological vector spaces and distributions,

vol. I, Addison-Wesley, Reading, Massachusetts, 1966.

576

J. MUJICA

4. ISIDRO, J.M.

Characterization of the spectrum o f some

topological algebras of holomorphic functions, these Proceedings,

5. MICHAEL, E.A.

Locally multiplicatively-convex topological

algebras, Memoirs Amer. Math. SOC., number 11, 1952.

6. MUJICA, J.

On the Nachbin topology in spaces of holomor-

phic functions, Bull, Amer. Math, soc. 81 (1975))

904-906.

7.

MUJICA, J.

Spaces of germs of holomorphic functions,

Advances in Math., to appear.

8. MUJICA, J.

Holomorphic germs on infinite dimensional

spaces, Infinite dimensional holomorphy and applicaticns, Notas de MatemGtica, North-Holland, Amsterdam,

1977,

~~.313-321* 9. NACHBIN, L. Topology on spaces of holomorphic mappings, Ergebnisse der Mathematik and ihrer Grenzgebiete, Band

47, Springer 10. NOVERRAZ, Ph.

Verlag, Berlin, 1969.

Pseudo-convexit6, convexit6 polynomiale et

domaines dlholomorphie en dimension infinie, Notas de Matemitica, North-Holland, Amsterdam, 1973. 11. SCHOTTENLOHER, M.

Polynomial approximation on compact

sets, Infinite dimensional holomorphy and applications, Notas de Matemgtica, North-Holland, Amsterdam,

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CAIXA POSTAL 1170 13.100 CAMPINAS, SP BRASIL

1977,