Complex homomorphisms of the algebras of holomorphic functions on Fréchet spaces

Math. Ann. 241, 73--82 (1979) Mathematischo Annalen ('c) by Spnnger-Verlag 1979

Complex Homomorphisms of the Algebras of Holomorphic Functions on Fr6chet Spaces

Jorge Mujica

[nstituto de Matemfitica, Universidade Estadual de Campinas. Caixa Postal 1170, BR-13100 Campinas SP, Brazil

I. Introduction

Let .~f(U) denote the algebra of all holomorphic functions on an open subset U of a complex locally convex space E. Let % denote the topology introduced independently by Coeur6 [2] and Nachbin [17], and which coincides with the bornological topology associated with the compact-open topology ro when the space E is metrizable. The topology ra seems to be the best suited for study of holomorphic continuation, as seen from the papers [2, 9, 3, 8, 22, 23, 1].

In this note we characterize the spectrum of (-~(U), z a) for certain domains in Fr6chet spaces. In a previous article [15] the author had shown that the spectrum of(.~(U),%) can be canonically identified with U when U is a polynomially convex domain in a Fr6chet space E with the approximation property. In this paper we prove that the same conclusion is true for U4°(U), r~) under the stronger assump- tion that the Fr6chet space E has the Banach approximation property. As a consequence we get that, when U is a Runge domain in a Fr6chet space with the Banach approximation property, then the spectrum of (.~(U), %) can be canonically identified with the envelope of holomorphy of U. We indicate briefly the organization of this article.

In Sect. 2 we extend the classical Banach-Dieudonn6 theorem concerning the dual E' of a metrizable locally convex space E to the spaces :~("E) of continuous m-homogeneous polynomials on E. The extension is straightforward, but the result is an important tool in the subsequent sections.

In Sect. 3 we characterize the spectrum of (~(U), r6) for certain domains in separable Fr6chet spaces with the approximation property. The results obtained apply only to a restricted class of domains, but the proofs already suggest how to proceed in more favorable situations.

In Sect. 4 we get our best results when we restrict our investigations, first to Fr6chet spaces with Schauder bases, and then pass to Fr6chet spaces with the Banach approximation property, using a well known result of Pelczynski [21]. As has become typical in this kind of situations, we deal first with Fr6chet spaces

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74 J. Mujica

having a continuous norm, and then pass to the general case using Dineen's surjective limits [6].

Throughout Sects. 3 and 4 one can notice the close connections between the study of the spectrum and the Levi problem, and also with problems on polynomial approximation. For example, as a byproduct of our methods, we obtain an extension of a theorem of Matyszczyk on polynomial approximation [14],

We refer to Nachbin [16] and Noverraz [19] for the definitions and properties of holomorphic functions on locally convex spaces. We follow the notation of [16].

2. Banach-Dieudonn6 Theorem for Polynomials

Let K P., and ~ denote, respectively, the set of positive integers, real numbers and complex numbers. If f is a complex-valued function defined on a set X we let

11 ftl x = sup tf(x)l • xEX

For each m~IN let ~'("E) denote the vector space of all continuous m-homogeneous polynomials on a locally convex space E. Throughout this paper all locally convex spaces will be assumed complex and Hausdorff. It follows from general topology that the compact-open topology z 0 coincides with the topology of simple convergence z s on each equicontinuous subset of ~(mE). I fE is metrizable we can say more.

2.1. Theorem. I f E is a metrizable locally convex space, then, for any ms N, the compact-open topology is the finest topology on f~(mE) which coincides with the topology of simple convergence on each equicontinuous subset of ~("E).

This is a Banach-Dieudonn6 theorem for homogeneous polynomials. The proof is similar to the classical proof of Dieudonn6 in the linear case: see [10, p. 245, Theorem 1]. Let z: denote the finest topology on ~(ZE) which coincides with z, on each equicontinuous subset. Clearly z: exists, is translation-invariant and is finer than %. Thus to prove the theorem it suffices to show that each open neighborhood of zero in ~("E) for vf is necessarily a neighborhood of zero for %. The critical step in the proof is the following lemma.

2.2. Lemma. Let E be a metrizable locally convex space, let ( ~ ) , ~ be a decreasing O-neighborhood base in E with V 1 = E and let U be an open neighborhood of zero in ~(mE) for zf. Then for each ns IN there exists a finite set A,C V, such that

(i~. Ai) ° nV"+ I C U

for every ns IN, where for any X C E,

X ° = {P~("E): t]P]I x < 11.

Holomorphic Functions on Fr6chet Spaces 75

Each set V ° is clearly equicontinuous and hence compact for T~ or %, by Ascoli

theorem. It is also clear that ~ ( " E ) = U V°. After these remarks, the proof of n~lq

Lemma 2.2 and that of Theorem 2.1 can proceed exactly as in the linear case.

3. Holomorphic Functions on Separable Fr6chet Spaces with the Approximation Property

Let iF(U) denote the algebra of all holomorphic functions on an open subset U of a locally convex space E. Let ~(E) denote the algebra of all continuous polynomials on E. Let ~s(E) denote the algebra of all continuous polynomials of finite type on E, i.e. the subalgebra of ~(E) generated by E'. Finally let ~I(mE)

A locally convex space E has the approximation property if for each compact set K C E and each 0-neighborhood V C E there exists a continuous linear operator of finite rank u:E~E such that u ( x ) - x e V for every .\'e K. I fE is a locally convex space with the approximation property, then it follows readily that ~'I(E) is dense in (~'(E),~0) and that ~'.("E) is dense in (~'("E),~0) for every m e N ; see [15, Corollary 3.5].

Let U be an open subset of a locally convex space E. For each countable open cover ~ of U we define

J'f~(~U)={fe~(U):{lfl}v<OO for all Ve'U}.

Then iF (U) is a Fr6chet space for the topology of uniform convergence on the sets VeU. The topology z~ is defined as the locally convex inductive topology on iF(U) with respect to the inclusion mappings ~¢F(U)~J'cF(U). When E is metrizable, z~ coincides with the bornological topology associated with z 0 on dr(U) (see [2] and [17]). A seminorm p on ~ ( U ) is T~-continuous if and only if, for each countable open cover "U of U, there exist V, . . . . . V,e ¢/~ and c >0 such that, if we

write V= U V/then i<_n

P(f)<cJ[f][v

for every fe~zg(U) (see [3, Lemma 7]).

3.1. Theorem. Let E be a separable Fr~chet space with the approximation property and let U be a balanced, convex open subset of E. Then the spectrum of (~ff(U), z~) can be canonically identified with U, i.e. for each continuous complex homomorphism T of (~(U),z~) there exists a unique point ~e U such that T(f)=f(~) for every f e,c~(u).

To prove the theorem we need the following lemma.

3.2. Lemma. Let E be a separable Frdchet space with the approximation property, let U be an open subset of E, and let T be a continuous complex homomorphism of (~ut~(U), z~). Then :

a) The restriction of T to each equicontinuous subset of 9f~(U) is continuous for the topology of simple convergence.

76 J. Mujica

b) For every me t~ the restriction of T to ~(mE) is continuous for the compact- open topology.

c) There exists a unique point ( e E such that T(P)= P(~) for every Pc ~(E).

Proof. (a) Let ~ be an equicontinuous subset of ~,~(U) and let e >0. Then for each x~ U there exists an open neighborhood V~ of x in U such that

tf(Y) - f(x)t < e (1)

for every f ~ and ye V~. Since E is separable and metrizable, the open cover (V~)x~ v of U admits a countable subcover. And since f -~[T(f) f defines a continuous seminorm on (~(U) , z~) there exists a finite set A = {x I . . . . . x,} C U and c > 0 such that, if we write V -- V~I w,. . w Vx~ then { T(f){ __< c If f Jr v for every f ~ ~,~(U). Substituting f~ for f , taking m-th root and letting m - ~ we get that

iT(f)] =< Ilfti v (2)

for every f ~ 3 f ( U ) . It follows from (1) and (2) that IT(f)] <= 1]fl]A+~ for every f ~ f f . Applying the previous argument to i f - f f we see that for each e > 0 there exists a finite set A C U such that

I T ( f ) - T(g)t < ]tf -gNA +~,

for every f and g in Y. Hence the restriction of T to f f is uniformly continuous for

~'s" (b) Follows from (a) and Theorem 2.1. (c) From (b) and the Mackey-Arens theorem [10, p. 205, Theorem 1] we see

that there exists a unique point ~ E such that T(40 = qS(~) for every ~bE E'. Hence T(P)=P(~) for every Pa~r(E). And since .~¢("E) is dense in (~(mE), Zo) (b) implies that T(P)= P(~) for every P ~ ( " E ) . The conclusion follows.

The next step in the proof of Theorem 3.1 is to show that the point ~ given by Lemma 3.2 lies in U. As pointed out to the author by Isidro, we could do this using [11, Lemmas 1 and 2], but we choose another method, which does not use the hypothesis that U is balanced, and is better suited for generalizations, as we shall see later.

For any set X in a locally convex space E we define

f 2 ~ = { x s E :]P(x)I-_<I[PIIx for all P~,C~(E)}.

3,3, Lemma. Let E be a locally convex space and let U be a convex open subset of E. Then for each point 4 ¢ U there exists an increasing sequence of open subsets V~ which cover U and such that ~¢(~)e for every j.

Proof. By the Hahn-Banach theorem there exists qSe E' such that Req~(x) < Req~(~) for every xe U. For each j e N let

Vj = {xe U : ]~b(x)[ < j , Re ~b(x) < Re~b(~) - l/j}.

Then the open sets Vj are increasing and cover U. Moreover, for each j, qS(Vj) is bounded and is bounded away from the line x = Re4~(~). Hence there exists an open disc D((j;Rj) such that qS(Vj)CD((j;Rj) and qS(~)¢b((j;Ri). We define


P~E ~(E) by P~(x) = ~b(x) - ~j for x~ E. Then

IIPjIIvj<R~, IP2(~)[ > R i •

Thus ~( I~) E for every j.

Proof of Theorem 3.1. Let T~ Spec(~'~(U), z~). By Lemma 3.2 there exists a unique point ~ E such that T(P)=P(~) for every P ~ ( E ) . Let (V~)2~ be any increasing sequence of open subsets of U which cover U. Since f ~ lZ(f ) l defines a continuous seminorm on (~(((U), %) there exists j~ IN and c > 0 such that IT(f)] <cllfl]vi for every fE~,~(U). As we did before, we may assume that c = 1. Then we get that

IP(~)[ = t T(P)I < [I Ptl v2

for every P ~ ( E ) . Thus ~(I~)~ and Lemma 3.3 implies that ~ U. Since U is balanced, ~(E) is dense in (~f((U), r~) by [4, Proposition 1.2], and it follows that T(f )=f(¢) for every f ~ W(U).

The proposition below will be useful in Sect. 4. Its proof is similar to that of Theorem 3.1 and may be omitted.

3.4. Proposition. Let E be a separable Frkchet space with the approximation property and let U be an open subset of E which can be written as the union of an increasing sequence of open subsets Vj such that (I~) E C U for every j. Then, for each continuous complex homomorphism T of (ovf(U), za), there exists a unique point ~ ~ U such that T(P)= P(~) for every P ~ ( E ) .

3.5. Example. Let U be a polynomial polyhedron in E, i.e. an open set of the form

U={xEE:IPi(x)I<I, i = l , . . . , n } ,

where P~E ~(E) for i = 1 . . . . . n. Then U satisfies the hypotheses of Proposition 3.4. It suffices to define

Vj={x~E: lPi (x ) l<l -1 / j , i=1 . . . . . n}.

3.6. Proposition. Let E be a locally convex space and let U be an open set in E with the property that for each T~Spec(~(U),zo) there exists a point ~ U such that T(~b)=qS(~) for every c~E E'. Then:

(a) For each sequence (x,) in U which converges to a point XoeC~U there exists an f ~ ~f(U) such that the sequence (f(x,)) does not converge.

(b) U is an open set of holomorphy.

Proof. (a) Suppose that (f(x,)) converges for each f~Juf(U). Then we define a complex homomorphism T on ~¢f(U) by T(f)=limf(x,) . We also define a seminorm p on ~(((U) by p(f)= sup If(x,)l. Then p is lower-semicontinuous for r o and hence for r~ and therefore continuous for r~ since (Jug(U), zo) is barrelled. It follows that T is z~-continuous. Then by hypothesis there exists a point ¢~ U such that T(q~)=~b(~) for every qS~E'. Then

qS(Xo) = lim ~b(x,) = T(~b)= ~b(~)

for every ~b~E' and hence Xo=¢eU, a contradiction. Therefore the sequence (f(x,)) does not converge for some f e ~ ( U ) .

The proof that (a)~(b) is standard and will be omitted.

78 J, Mujica

4. Holomorphic Functions on Fr~het Spaces with the Banach Approximation Property

A Fr6chet space E has the Banach (or bounded) approximation property if the identity operator is the pointwise limit of a sequence of continuous linear operators of finite rank. Every Fr6chet space with the Banach approximation property is separable and has the approximation property.

A sequence (e,) in a Fr~chet space E is a Schauder basis if each x~E admits a unique representation x= ~ x . e . with x , ~ . A Fr6chet space with a Schauder basis has the Banach approximation property. Conversely, Pelczynski has proved that every Fr6chet space with the Banach approximation property is a complemented subspace of a Fr~chet space with a Schauder basis; see [21] or [14, Theorem 2.11 ].

An open set U in a Fr6chet space E is polynomially convex i f / (E~ U is compact for every compact set K C U.

4.1. Theorem. Let E be a Fr~chet space with the Banach approximation property and let U be a connected polynomially convex open set in E. Then the spectrum of (~(U), ~) can be canonically identified with U.

Before proving the theorem we will give two lemmas. The main ingredients of Lemma 4.2 below have already appeared in the proofs of several results ; see [8, Theorem 1; 19, p. 83, Theorem 5.2.1; 5, Proposition 1.2; 13, Theorem 1, and 14, Theorem 2.2].

4.2, Lemma. Let E be a Fr~chet space with a Schauder basis and having a continuous norm. Let U be a potynomially convex open subset of E. Then U can be written as the union of an increasing sequence of open subsets V~ such that, for each j, (~)E is contained in U and bounded away from OU.

Proof. Let (e,) be a Schauder basis for E, let E, denote the subspace generated by el, ..., % and let E~o = uE, . Let n, : E ~ E , denote the canonical projection, and choose an increasing sequence of seminorms F, generating the topology of E, and such that

0~(x) = sup0~(n,(x))

for every ~ F and x~E. It is easy to see that U can be written as a countable union of open balls

B,,(x~; r~), where 0q~ F, xie Eoo and r~ > 0 is such that B~,(xi; 2r~) C U. For each j~ N let

Vj = [_j B~,(x/; ri), /3j = max ~i, sj = min r i . i<=j- • i < _ j i<=j

Then the sets Vj are open and increasing and cover U. We also have that

Vi+ B~j(O;sj)C U (1)

and we will show that

(~)e + Ba~(O ; s)) C U. (2)

This will be done in several steps.


i) We claim that for each j there exists nj such that

g,(b) = Vf~E, (3)

for every n>nj. The proof is easy: just choose nj such that xieE. , for i= 1 . . . . . j. ii) Write U, = UrgE, and V~, = Vj~E,. Then using (3) one can easily prove that

for every j /".- .~x

0 (4) n>_nj

iii) We claim that Vj. is relatively compact in U n for every) and every n. Since E has a continuous norm we may assume that each ~ e F is a norm. Let ~o be the smallest member of F. Then each Vj is %-bounded and so is each ~ . Hence Vj, is a bounded subset of E~. On the other hand from (1) we get that

Vj~ + Bpj(O;sj)c~E n ( Un (5)

and Vjn is bounded away from ~3U~. iv) Finally, to show (2) it will be sufficient to show that whenever 0 < t~ <sj then

(~)E + Bp,(0; t j) ( U. (6)

Since Vj, is relatively compact in U, and since Un is a polynomially convex open subset of E,, (5) implies that

(Vj,)E" + B~,(0 ; sj)c~ E, C U,. (7)

Let us fix tj, 0<t~<sj , and define

Wj = {xe U :di(x, E\U) > s i - t j},

where d~(x,E\U)=inff l j(x-y). We also set Wi, = W/~E,. Then from (7) we get that YCU

+ ;, t )nE, c (8)

Suppose that (6) is not true. Then we can find ~e(~)E and r/¢U with flj(~-r/)<tj. Then q~14f and since ~). is closed in E and n,(q)~q we conclude that n,(q)¢W i for all n sufficiently large. On the other hand, (4) implies that n,(~)e(Vj,)E" for all n sufficiently large. But since fli(n,(~)- n.(r/) < tj we are contradicting (8). This shows (6) and the lemma.

4.3. Lemma. Let E be a Frdchet space with a Schauder basis, let U be a connected polynomially convex open set in E, and let f =(~j) be an increasing sequence of seminorms generating the topology of E. Then

(a) For each) the quotient space F j = E/~f 1(0) is a FrOchet space with a Schauder basis and has a continuous norm.

(b) I f aj:E-*Fj denotes the quotient mapping then there exists Jo such that, for every j>jo, U =a] l(crj(U)), and aj(U) is a potynomially convex open set in Fj.

(c) For each f ¢ oZd~(U) there exists j > jo and an f ie Jt°(aj(U)) such that f = f ~,:,a j on U.

For a proof of (a) see [6,Example 2.4] or [20, Lemma 3]. For a proof of (b) see [5, Lemmas 1.1 and 1.2] or [19, p. 43, Theorem 2.1.7]. The proof of (c) is also standard; see [18, Proposition 1].

80 J. Mujica

Proof of Theorem 4.1. i) First assume that E has a Schauder basis and a continuous norm. Then given TeSpec(Jg(U),ro), by Lemma4.2 and Proposition 3.4 there exists a unique point ~e U such that T(P)=P(~) for every Pe~(E). Now, by [14, Theorem 2.9], ~@(E) is sequentially dense in ~(U) , i.e. for each fs,;4~(U) there exists a sequence (P~)C~(E) which converges to f in ()¢'(U),r6). And since by Lemma 3.2 the restriction of T to each equicontinuous subset of .~(U) is %-continuous, we get that T( f )= f (O for every fe;.gg(U).

ii) If we only assume that E has a Schauder basis, then the reduction to case i), using Lemma 4.3, is standard (see [6]).

iii) In the general case, the reduction to case ii), using Pelczynski's result [21], is also standard (see [19]).

4.4. Example. In [12] Josefson gives an example of a pseudoconvex domain U in E=co(A ), with A uncountable, which is not a domain of holomorphy. As Schottenloher points out in [25] U is even polynomially convex. Josefson exhibits a domain U 1 in E which contains U properly and such that each fEJg(U) extends to an f~.~f(U1). Then by [9, Theorem5.8], for each ~EU~ the mapping f ~ ~(U)-.f~(O~ff2 defines an element in Spec (0~(U), %). Josefson exhibits a point ~sUI\U such that the corresponding homomorphism f~f~(~) is not Zo-Continuous. Actually, the same conclusion is true for any ~eUI\U for, according to [15, Theorem 5.2], Spec(o;~{U),%) can be canonically identified with U. In particular, U is a polynomially convex domain in a Banach space with the approximation property where Spec (,~(U), %) # Spec (H(U), z~).

An open set U in a locally convex space E is said to be Runge if ~(E) is dense in (~(U),v0). We say that U is sequentially Runge if ~(E) is sequentially dense in (~(U), %). Then the result below can be easily obtained from Theorem 4.1 using [19, p. 78, Theorem 4.4.1].

4.5. Theorem. Let E be a Frdchet space with the Banach approximation property and let U be a connected Runge open set in E. Then the spectrum of (j~F(U), za) can be canonically identified with the envelope of' holomorphy of U.

4.6. Corollary. Let E be a FrOchet space with the Banach approximation property and let U be a connected Runge open set in E. Then U is a domain of holomorphy if and only if the spectrum of (Jg~(U), z~) can be canonically identified with U.

Theorem4.5 solves partially a problem posed by Schottenloher [24, Problem 6]. Corollary 4.6 extends partially a result of Aurich [1, Theorem 3.3].

We conclude this article with a result on polynomial approximation, which extends slightly a result of Matyszczyk [t4, Theorem 2.12].

4.7. Theorem. Let E be a Frdchet space with the Banach approximation property and let U be a connected potynomially convex open set in E. Then U is sequentially Runge.

tn the case where E has a Schauder basis and a continuous norm, this is nothing but [14, Theorem 2.9]. Then the reduction to the general case using Lemma 4.3 and Pelczynski's result [21] is standard (see [6] and [t9]).


W e h a v e r e s t r i c t e d o u r i n v e s t i g a t i o n s to F r 6 c h e t spaces , b u t t he r e su l t s s h o u l d

a lso h o l d in o t h e r s u i t a b l e c lasses o f l oca l ly c o n v e s spaces . In a p r i v a t e c o m -

m u n i c a t i o n , S. D i n e e n h a s p o i n t e d o u t t o t h e a u t h o r t h a t t he s p e c t r u m of

(Yd'(U), %) c a n b e c a n o n i c a l l y iden t i f i ed w i t h U, w h e n U is a n o p e n p o l y d i s c in a

fully n u c l e a r s p a c e w i t h a S c h a u d e r b a s i s (see [ 7 ] for t he t e r m i n o l o g y ) .

Acknowledgement. I would like to thank both J. M. Isidro and the referee for useful suggestions, in particular for suggesting the present version of Theorem 3.1. I would also like to thank Ph. Boland for pointing out an inaccuracy in the original manuscript.

References l. Aurich, V. : The spectrum as envelope of holomorphy of a domain over an arbitrary product of

complex lines. In : Proceedings on infinite dimensional holomorphy. Lecture Notes in Math. 364, pp. 109 122. Berlin, Heidelberg, New York: Springer 1974

2. Coeur6, G. : Fonctions plurisousharmoniques sur les espaces vectoriels topologiques et applications a l'6tude des fonctions analytiques. Ann. Inst. Fourier (Grenoble) 20, 361 432 (1970)

3. Dineen, S. : The Cartan-Thullen theorem for Banach spaces. Ann. Scuola Norm. Sup. Pisa 24, 667- 676 (1970)

4. Dineen, S. : Holomorphic functions on locally convex topological vector spaces, I. Locally convex topologies on X(U). Ann. Inst. Fourier (Grenoble) 23, 19-54 (1973)

5. Dineen, S. : Holomorphic functions on locally convex topological vector spaces. II. Pseudo convex domains. Ann. Inst. Fourier (Grenoble) 23, 155-185 (1973)

6. Dineen, S. : Surjective limits of locally convex spaces and their applications to infinite dimensional holomorphy. Bull. Soc. Math. France 1103, 441--509 (1975)

7. Dineen, S.: Analytic functionals on fully nuclear spaces (preprint) 8. Dineen, S., Hirschowitz, A. : Sur le th6or~me de Levi banac, hique. C.R. Acad. Sc. Paris 272, 1245-

1247 (1971) 9. Hirschowitz, A. : Prolongement analytique en dimension infinie. Ann. Inst. Fourier (Grenoble) 22,

255-292 (1972) 10. Horv~tth, J. : Topological vector spaces and distributions. I. Reading, Mass. : Addison-Wesley 1966 II. Isidro, J.M.: Characterization of the spectrum of some topological algebras of holomorphic

functions. In: Advances in holomorphy, pp. 407--4t6. Amsterdam: North-Holland 1979 IZ Josefson, B.: A counterexample in the Levi problem. In: Proceedings on infinite dimensional

holomorphy. Lecture Notes in Math. 364, pp. 168-177. Berlin, Heidelberg, New York: Springer 1974

13. Matyszczyk, C. : Approximation of analytic operators by polynomials in complex Bo-spaces with bounded approximation property~ Bull. Acad. Polon. Sci. 20, 833-836 (1972)

14. Matyszczyk, C. : Approximation of analytic and continuous mappings by polynomials in Fr6chet spaces. Studia Math. 60, 223-238 (1977)

15. Mujica, J.: Ideals of holomorphic functions on Fr6chet spaces, tn: Advances Jn holomorphy. pp. 563 576. Amsterdam: North-Holland 1979

t6. Nachbin, L. : Topology on spaces of holomorphic mappings. Berlin, Heidelberg, New York: Springer 1969

17. Nachbin, L. : Sur les espaces vectoriels topologiques d'applications continues. C.R. Acad. Sc. Paris 27l, 596-.-598 (1970)

18. Nachbin, L.: Uniformit6 d'hotomorphie et type exponential. In : S6minaire Pierre Lelong ann6e 1970. Lecture Notes in Mathematics 205, pp, 216-224. Berlin, Heidelberg, New York: Springer 1971

19. Noverraz, Ph. : Pseudo-convexit6, convexit6 polynomiale et domaines d'holomorphie en dimension infinie. Amsterdam: North-Holland 1973

20. Noverraz, Ph. : Pseudo-convexit6 et base de Schauder dans les elc. In : S6minaire Pierre Lelong ann6e 1973/74. Lecture Notes in Mathematics 476, pp. 63-82. Berlin, Heidelberg, New York: Springer 1975

21. Pelczynski, A.: Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis. Studia Math. 40, 239-243 (1971)

82 J. Mujica

22. Schottenloher, M.: fiber analytische Fortsetzung in Banachr~ume. Math. Ann. 199, 313-336 (1972)

23. Schottenloher, M. : Analytic continuation and regular classes in locally convex Hausdorff spaces. Portug. Math. 33, 219-250 (1974)

24. Schottenloher, M.: Riemann domains: basic results and open problems. In: Proceedings on infinite dimensional holomorphy. Lecture Notes in Mathematics 364, pp. 196-212. Berlin, Heidelberg, New York: Springer 1974

25. Schottenloher, M. : Polynomial approximation on compact sets. In: Infinite dimensional holomorphy and applications, pp. 379-391. Amsterdam: North-Holland 1977

Received June 16, and in revised form October 14, 1978

Complex homomorphisms of the algebras of holomorphic functions on Fréchet spaces

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