STRUCTURE OF SPACES OF GERMS OF HOLOMORPHIC FUNCTIONS

Let E be a Frechet (resp. Frechet-Hilbert) space. It is shown that E ∈ (Ω) (resp. E ∈ (DN)) if and only if [H(OE)]′ ∈ (Ω) (resp. [H(OE)]′ ∈ (DN)). Moreover it is also shown that E ∈ (DN) if and only if Hb(E′) ∈ (DN). In the nuclear case these results were proved by Meise and Vogt [2]. 1. Preliminaries 1.1. Let K be a compact set in a Frechet space E. By H(K) we denote the space of germs of holomorphic functions on K. This space is equipped with the inductive topology H(K) = lim ind U↓K H∞(U). Here for each neighborhood U of K, by H∞(U) we denote the Banach space of bounded holomorphic functions on U with the sup-norm ‖f‖U = sup{|f(z) : z ∈ U}. 1.2. Let E′ denote the strong dual space of a Frechet space E. A holomorphic function on E′ is said to be of bounded type if it is bounded on every bounded set in E′. By Hb(E′) we denote the metric locally convex space of entire functions of bounded type on E′ equipped with the topology the convergence on bounded sets in E′. For more details concerning holomorphic functions on locally convex spaces we refer to the book of Dineen [1]. 468 N. Van Khue, P. Thien Danh 1.3. Assume the topology of E is defined by an increasing fundamental system of seminorms {‖ · ‖k}k=1. For each subset B of E, define the generalized seminorm ‖ · ‖B : E′ → [0,+∞], by ‖u‖B = sup{|u(x)| : x ∈ B}. Write ‖ · ‖q for B = Uq = {x ∈ E : ‖x‖q ≤ 1}. Using this notation define E to have the property (DN) : ∃ p∀ q ∃ k, C > 0 : ‖ ‖q ≤ C‖ ‖k‖ ‖p. (Ω) : ∀ p∃ q ∀ k ∃ d, C > 0 : ‖ · ‖∗1+d q ≤ C‖ · ‖k‖ · ‖∗d p . The properties (DN), (Ω) and the other many properties were introduced and investigated by Vogt (see, for example, [7], [8], etc.). In [8] he has proved that E ∈ (DN) (resp. E ∈ (Ω)) if and only if E is isomorphic to a subspace (a quotient space) of the space B⊗̂πs for some Banach space B, where s is the space of rapidly decreasing sequences. The following three theorems are proved in the present paper. Theorem 1. Let E be a Frechet space. Then the following are equivalent (i) E ∈ (Ω) (ii) [H(K)]′ ∈ (Ω) for some non-empty compact set K in E. (iii) [H(K)]′ ∈ (Ω) for all compact sets K in E. Theorem 2. Let E be a Frechet-Hilbert space. Then E ∈ (DN) if and only if [H(OE)]′ ∈ (DN). Theorem 3. Let E be a Frechet space. Then E ∈ (DN) if and only if Hb(E′) ∈ (DN). The proofs of Theorems 1, 2 and 3 are presented in Sections 2, 3 and 4 respectively. Structure of spaces of germs of holomorphic functions 469 2. Proof of Theorem 1 2.1. Lemma. Let E be a Frechet space. Then E ∈ (Ω) if and only if E′ is isomorphic to a subspace of B⊗̂πs′ for some Banach space B. Proof: Suppose E′ is isomorphic to a subspace of the space B⊗̂πs′ where B is some Banach space. Then E′′ is isomorphic to a quotient space of (B⊗̂πs′)′ ∼= B′⊗̂πs and hence E′′ ∈ (Ω). This implies that E ∈ (Ω) since ‖u‖k = sup{|v(u)| : v ∈ U k ⊂ E′′} for u ∈ E′. Conversely assume that E ∈ (Ω). Consider the canonical resolution 0 −→ E −→ ∏ k≥1 Ek R −→ ∏ k≥1 Ek −→ 0 as constructed by Palamodov [4], where for each k ≥ 1, Ek stands for the Banach space associated to ‖ ‖k. Since E is isomorphic to a quotient space of B⊗̂πs with B is some Banach space [8], E is quasinormable. Hence we may assume that every bounded set in Ek+1 can be approximated by a bounded set in Ek+2 under the canonical map Ek+2 → Ek. It follows from [4] that every bounded set in ∏ k≥1 Ek is the image of a bounded set in ∏ k≥1 Ek under R. By modifying the argument in [8] we imply that E is isomorphic to a quotient space of B⊗̂πs for which E′ is isomorphic to a suspace of (B⊗̂πs)′ ∼= B′⊗̂πs′. The following lemma is an immediate consequence of the preceding lemma. 2.2. Lemma. Let E be a Frechet space. Then E ∈ (Ω) if and only if E′′ ∈ (Ω). 2.3. Lemma. Let B be a Banach space. Then [H(OB⊗̂πs)] ′ ∈ (Ω). Proof: Let {ej} be the canonical basis of s with the dual basis {ej} of s′. Since s is nuclear without loss of generality we may assume that 470 N. Van Khue, P. Thien Danh δp = ∑ j≥1 ‖ej‖p+1‖ej‖p < 1/e for p ≥ 1.


Introduction
Let E be a Fréchet space and let K be a compact subset in E. By H(K) we denote the space of germs of holomorphic functions on K equipped with the inductive limit topology.Some linear topological invariants, in particular those of the (Ω)-type for the strong dual [H(K)] of the space H(K), were investigated by several authors.For example, in the finite dimensional case, Zaharjuta proved that [H(K)] has ( Ω) if and only if K is L-regular [17].This problem, in the infinite dimensional case, has been considered already by R. Meise, D. Vogt and many others.Meise and Vogt have shown in [7] that [H(K)] has (Ω) for every compact subset K in a nuclear Fréchet space E as long as E has (Ω).Recently, this result has been extended to the general case where E is only Fréchet by Nguyen Van Khue and Phan Thien Danh [10].For the invariants ( Ω) and ( Ω), Meise and Vogt in [8] gave some necessary and sufficient conditions for the compact polydiscs D in a nuclear Fréchet space having a Schauder basis such that [H( D)] has ( Ω) and has ( Ω) respectively.
The aim of the present paper is to study the invariant (LB ∞ ) as well as ( Ω) and ( Ω) of [H(K)] in the case where K is a balanced convex compact subset of a nuclear Fréchet space E. It should be mentioned that this problem has been treated very recently by Le Mau Hai and Nguyen Van Khue [6] in the case where E is a Fréchet-Schwartz space having an absolute basis.Our main results are explained in Sections 2 and 3. Namely, in Section 2 by employing an important characterization of (LB ∞ ) for Fréchet spaces [15], we prove that if B is a balanced convex compact subset of a Fréchet space E having ( ΩB ) then [H(B)] has (LB ∞ ) (Theorem 2.1).In Theorem 2.2, under the additional assumption that E has the bounded approximation property, we prove that B is not pluripolar if [H(B)] has (LB ∞ ).Combining this result and a characterization of ( ΩB ) in terms of the non-pluripolarity of B [2] we also obtain a converse to Theorem 2.1 in the special case mentioned above.In Section 3, we prove in Theorem 3.1 that if B is a balanced compact subset of a nuclear Fréchet space having a Schauder basis then [H(B)] has either ( ΩB ) or ( ΩB ) if and only if E has the same property.
Finally, we note that the invariants of (DN )-type for spaces of entire functions of bounded type on (DF )-spaces were considered by several authors (for example [6], [10], . . .).

Preliminaries
1.1.Some linear topological invariants.Let E be a Fréchet space with a fundamental system of semi-norms Using this notation we say E has the property The above properties were introduced and investigated by Vogt (see [9] or [16] for (Ω) and [15] for the others).

Holomorphic functions. Let E, F be locally convex spaces and D an open subset in
we denote the space of F -valued holomorphic functions on D, equipped with the compact-open topology.When F is omitted, it is understood to be the scalar field C, e.g.H(D) = H(D, C).
Finally for each compact set K in E, by H(K) we denote the space of holomorphic functions on K, equipped with the inductive topology, i.e.

H(K)
where U ranges over all neighbourhoods of K and H ∞ (U ) denotes the Banach space of bounded holomorphic functions on U .
For the details concerning the holomorphic functions and the germs of holomorphic functions on compact sets in a locally convex space, we refer to the book of Dineen [1].

The structure (LB ∞ )
Theorem 2.1.Let E be a nuclear Fréchet space and B a balanced convex compact subset in E. Assume that E has ( ΩB ): Then Note that in the definition of ( ΩB ), by choosing q sufficiently large, we may assume that C = 1.
We need the following: Lemma 2.2.Let E and B be as in Theorem 2.1.Then B is a set of uniqueness.
Here we say that the compact set B is a set of uniqueness if for every Since P n f are n-homogeneous polynomials and P n f |B = 0, it follows that P n f |span B = 0.By the continuity of P n f and by span B = E, we have P n f = 0 for n ≥ 0. Thus f = 0 in W and hence B is a set of uniqueness.

Proof of Theorem 2.1: Since
where α = (α j ) with α j = j for j ≥ 1, by Vogt's theorem it suffices to show that every continuous linear map . By Grothendieck's factorization theorem [9], this yields that f : Obviously g is separately holomorphic in the sense of Sciak [14], this means that g(x, •) is holomorphic in λ ∈ C for every x ∈ B and g(•, λ) is too in x ∈ W for every λ ∈ ∆.We denote by F the family of all finite dimensional subspaces P = 0 of E(B), where E(B) is the Banach space spanned by B. For each P ∈ F consider g P = g |((B∩P )×C )∪((W ∩P )× ∆) .Since B ∩ P is the unit ball in P and ∆ is not polar, by Nguyen Thanh Van-Zeriahi [11] g P is uniquely extended to a holomorphic function gP on (W ∩ P ) × C. The uniqueness implies that the family {g P : P ∈ F} defines a Gâteaux holomorphic function g on (W ∩ E(B)) × C. On the other hand, since g is holomorphic on (W ∩ E(B)) × ∆, Zorn's theorem [1] implies that g is holomorphic on (W ∩ E(B)) × C. Consider the holomorphic function ĝ : (W ∩ E(B)) −→ H(C) associated to g.We prove that ĝ can be extended to a bounded holomorphic function on a neighbourhood of B with values in H(C).
(ii) The following is a modification of Meise-Vogt [8] and of Le Mau Hai [5].
be two fundamental systems of seminorms of E and H(C) respectively.Since H(C) has (DN ) we have Note that by replacing k with some k > k, we always may assume that C = 1.Choose α such that U α ⊂ W and Let ω α from E into E α , the Banach space associated to • α , be the canonical map and without loss of generality we may assume that E(B) and E α are Hilbert spaces.
Then, by [12, Proposition 8.6.6,p. 143], A can be written in the form where λ j > 0 ∀ j ≥ 1, λ = (λ j ) ∈ s, the space of rapidly decreasing sequences, (y j ) is a complete orthonormal system in E(B) and (z j ) an orthonormal system in E α .Since we have It follows that where

N. Dinh Lan
We set Then Now put and choose β such that For β sufficiently large, we can choose C = 1.From ( 1)-( 3) we have where Ûα is the unit ball in E α .This may be illustrated in the following diagram. where From the relation we deduce that ĥ   On the other hand, by Cauchy's theorem, we get It follows that where We have where Since H(C) has (DN ), for every q ≥ p and d = d δ there exists k ≥ q and C > 0 such that Again we may assume C = 1.Then Since λ = (λ j ) ∈ s, the sequence We have Hence the form (iii) Consider the separately holomorphic function h 1 in the sense of Siciak [14] on (δU β × C) ∪ (W × ∆), induced by ĥ1 and g.By the same argument as in (i), h 1 is holomorphically extended to a function h1 on W ×C. Let ĥ1 : W −→ H(C) denote the holomorphic function associated to h1 .Since B is convex, balanced and the equality ( ĥ1 − ĝ) δU β ∩B = 0 holds, from the Taylor expansion of ( ĥ1 − ĝ) (iv) Applying a similar argument as in (ii) to each point of W , it follows that ĥ1 is locally bounded.Thus, by shrinking W , without loss of generality, we may assume that ĥ1 (W ) is bounded.Define the continuous linear map

N. Dinh Lan
We have On the other hand, since B is of uniqueness and H(B) is reflexive, it follows that S = T .Hence T is compact.
For the formulation of the second theorem we recall the following [2], [3]: An upper-semicontinuous function ϕ :

Theorem 2.3. Let E be a nuclear Fréchet space with the bounded approximation property and B a balanced convex compact subset in E.
Then the following assertions are equivalent: For the proof of Theorem 2.3 we need the following lemma which was proved independently in [6].
Since f |K = 0 and K is compact, it follows that ε k 0. By applying the (LB ∞ ) property of [H(K)] to the sequence ρ k = √ − log ε k +∞ and to p = 1, we can find q ≥ 1, N 1 ≥ 1 and C > 0 such that This inequality gives Then Then Ω ϕ is pseudoconvex.Since E has the bounded approximation property, there exists f ∈ H(Ω ϕ ) such that Ω ϕ is the domain of existence of f (by [13]).Write the Hartogs expansion of f , It is easy to see that h n are holomorphic on E, because of the uppersemicontinuity of ϕ.
Let g : B −→ H(C) given by g(x)(λ) = f (x, λ) for x ∈ B, λ ∈ C. Then g is weakly holomorphic.Indeed, given µ ∈ [H(C)] , take r > 0 such that µ can be considered as a continuous linear functional on H ∞ (r∆).Since B × C ⊂ Ω ϕ , we can find a neighbourhood V of B in E such that V ×r∆ ⊂ Ω ϕ .Hence f induces a holomorphic extension of µ•g to V .On the other hand, since B is a set of uniqueness, the form µ, → µ • g, the unique holomorphic extension of µ • g for µ ∈ [H(C)] , defines a linear map T : [H(C)] −→ H(B).Again since B is a set of uniqueness, T has a closed graph.The closed graph Grothendieck theorem [4] yields that T is continuous.By Vogt [15] we can find a neighbourhood W of 0 ∈ [H(C)] such that T (W ) is bounded in H(B).By the regularity of H(B) [1] there exists a neighbourhood V of B in E such that T (W ) is contained and bounded in H ∞ (V ).This implies that g is extended to a holomorphic function ĝ : V −→ H(C).Obviously g = f on non-empty open subset of Ω ϕ , where g(x, λ) = ĝ(x)(λ) for x ∈ V , λ ∈ C. By the hypothesis Ω ϕ is the domain of existence of f , thus we have

Sufficiency. It suffices to prove the case E ∈ ( ΩB ).
Let (e j ) be a basis of E and (e * j ) its dual basis in E .Since E is nuclear, without loss of generality we may assume that j≥1 e * j * q+1 e j q < 1 e 2 for q ≥ 1.From the above inequalities, it follows also that H(B) ∼ = lim ind q H q ,

1 .
Let E be a nuclear Fréchet space with a basis and B a balanced compact subset in E. Then [H(B)] has either ( ΩB ) or ( ΩB ) if and only if E has the same property.Proof: Necessity.Since the forms f → f (0) and u → [u], where [u] denotes the element of H(B) induced by u ∈ E , define the continuous linear maps P : H(B) −→ E and Q : E −→ H(B) satisfying P • Q = id, it follows that E can be considered as a subspace of H(B).Hence E ∼ = E which is a quotient space of [H(B)] .This proves the necessity of the theorem.
∈ U q and x ∈ B.