Spaces of N-homogeneous polynomials between Fréchet spaces

J. Math. Anal. Appl. 297 (2004) 587–598 www.elsevier.com/locate/jmaa

Spaces of N-homogeneous polynomials between Fréchet spaces Elisabetta M. Mangino Dipartimento di Matematica “E. De Giorgi”, Università degli Studi di Lecce, I-73100 Lecce, Italy Received 20 January 2004

Submitted by R.M. Aron Al Professore Horváth, in occasione del suo ottantesimo compleanno

Abstract Properties of the Nachbin-ported topology on spaces of N-homogeneous polynomials between Fréchet spaces are investigated by applying results on derived functors.  2004 Elsevier Inc. All rights reserved. Keywords: N -homogeneous polynomials; Nachbin-ported topology; Functor Ext1

The space of Fréchet valued N -homogeneous polynomials on a Fréchet space can be represented as a space of Fréchet valued continuous linear maps on a symmetric projective tensor product. We investigate the topology that is induced by the Nachbin-ported topology τω on this space of linear maps, in order to get information about the properties of the space of N -homogeneous polynomials. This study is motivated by the analysis of topological properties of spaces of Fréchet valued holomorphic functions on balanced open subsets of Fréchet spaces, when endowed with the Nachbin-ported topology or the τδ -topology. While in the case of Banach-valued holomorphic functions defined on Fréchet spaces much is known (see, e.g., [5, Chapters 3, 4]), the case of Fréchet valued holomorphic functions seems to present still many open questions. In this note, we will mainly study when spaces of N -homogeneous polynomials (endowed with the topology τω ) between Fréchet spaces are bornological. The first section collects some preliminary definitions and results about projective and inductive spectra and about spaces of N -homogeneous polynomials. In the second section E-mail address: [email protected]. 0022-247X/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.03.045

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we study the Nachbin-ported topology on spaces of continuous linear functions between Fréchet spaces. In particular, we give a description of this topology that holds in many interesting cases and that permits us to apply the classical results about the vanishing of functor Ext1 to get information on topological properties. The results of this section are applied to spaces of N -homogeneous polynomials between Fréchet spaces, by linearizing these spaces, in Section 3. A series of examples involving topological invariants such as the properties (DN) and (Ω) are presented in Section 4. Finally, by combining results about the decomposition of spaces of holomorphic functions with the results of the third section, we present some consequences for spaces of holomorphic functions between Fréchet spaces.

1. Preliminaries If E is a locally convex space, we denote by cs(E) the set of all continuous seminorms on E, by U0 (E) the set of all 0-neighbourhoods of E and by B(E) the set of all bounded subsets of E. If p ∈ cs(E), then its dual norm is defined by p∗ (y) := sup{|y(x)|: x ∈ E, p(x)
1} for every y in the dual space E  . For any subset A of E, Γ (A) stands for the absolute convex hull of A. If X is a normed space, let BX denote its open unit ball. A Fréchet space is said to be a proper Fréchet space if it is not normable. If E and F are locally convex spaces, let L(E, F ) be the space of all linear and continuous maps between E and F and Lb (E, F ) the same space endowed with the topology of uniform convergence on bounded subsets of E. If A ⊆ E and B ⊆ F , set W (A, B) := {f ∈ L(E, F ): f (A) ⊆ B}. A (countable) projective spectrum is a sequence X = (Xn , ρn,k )n,k∈N, n
k of locally convex spaces Xn and linear and continuous maps ρn,k : Xk → Xn such that, for every n, k, h ∈ N, ρn,k ◦ ρk,h = ρn,h and ρn,n is the identity. In this case we set
  Proj (X ) := (xn )n∈N ∈ Xn : ∀n ∈ N, ρn,n+1 (xn+1 ) = xn , 0

Ψ:



Xn →

n∈N

Proj1 (X ) :=



n∈N

Xn ,

  (xn )n∈N → ρn,n+1 (xn+1 ) − xn n∈N ,

n∈N



Xn / Im Ψ.

n∈N

 The space Proj0 (X ), endowed with the topology which is induced by n∈N Xn , is denoted by projn (Xn , ρn,n+1 ) (or by projn Xn if the maps ρn,n+1 are clear in the context) and is called the projective limit of X . A projective limit projn (Xn , ρn,n+1 ) is said to be reduced if the canonical maps ρn : Proj0 (X ) → Xn , (xk )k∈N → xn have dense range for every n ∈ N. A sequence E = (En , jnm )n,m∈N,n
m of locally convex spaces En and linear continuous inclusion maps jnm : En → Em (n
m) is called an (injective countable) inductive spectrum. Let E be the union of the spaces En and endow it with the finest locally convex topology for which all the injections jn : En → E are continuous. Then E is called the inductive limit of the spectrum E and is denoted by indn En .

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By a result of Vogt [17] (see also [19, Theorem 3.3.4]), if X is a projective spectrum of separated (LB)-spaces (i.e., countable inductive limits of Banach spaces) such that Proj1 X = 0, then its projective limit is ultrabornological. For further notation and information on the general theory of locally convex spaces, we refer to [7–9,12]. In particular, we refer to [12] for Köthe echelon spaces λp (A) (1
p
∞) and power series spaces Λr (α) (r = 1, ∞). We refer to [5] for notation and results on N -homogeneous polynomials and holomorphic functions between complex locally convex spaces. We only recall the definition of the ported topology τω and of the τδ -topology. If U is an open subset of a locally convex space E and F is a normed space, we say that a seminorm p on the space of holomorphic functions H(U, F ) is ported by the compact subset K of U if for every V open, K ⊂ V ⊂ U , there exists c(V ) > 0 such that p(f )
C(V ) supx∈V f (x) for all f ∈ H(U, F ). The Nachbin-ported topology τω on H(U, F ) is the topology generated by all seminorms ported by all compact subsets of U . A seminorm p on H(U, F ) is said to be τδ -continuous if for each increasing countable open cover (Vn )n∈N of U there exist n0 ∈ N and C > 0 such that   p(f )
C sup f (x) for all f ∈ H(U, F ). x∈Vn0

The τδ topology on H(U, F ) is the topology generated by all τδ -continuous seminorms. When F is a complete locally convex space, the τω and τδ -topologies are defined by     H(U, F ), τ = projα∈cs(F ) H(U, Fα ), τ (τ = τω , τδ ), ˆ with α(x ˆ +ker α) = α(x). where, for every α ∈ cs(F ), Fα is the normed space (F / ker α, α), It is known that τω and τδ induce the same topology on the spaces P(N E, F ) of N -homogeneous polynomials that can be described as follows:  N     P E, F , τω = projβ∈cs(F ) indα∈cs(E) Pb N Eα , Fβ ; here the linking maps are the canonical ones given by the factorization lemma (see [5, Lemma 1.13, Proposition 1.15]) and Pb (N Eα , Fβ ) is the space of N -homogeneous polynomials from Eα into Fβ , endowed with the topology of uniform convergence on the bounded subsets of Eα . When N = 1, set Lω (E, F ) := (P(1 E, F ), τω ).

2. The space Lω (E, F ) If E and F are Fréchet spaces, under some assumptions, it is possible to give another description of the ported topology on P(N E, F ). To this aim, first we describe explicitly the topology τω in L(E, F ) and we introduce a topology τi . Let E be a metrizable space and (X,  · ) a normed space. Let (Un )n∈N be a decreasing fundamental sequence of 0-neighbourhoods in E, (pn )n∈N their Minkowski functionals. For every n ∈ N, set En := E/ker(pn ), ρn : E → En , x → x +ker(pn ) and ρnm : Em → En , x + ker(pm ) → x + ker(pn ), if m  n. Endow En with the norm pˆ n (x + ker(pn )) := pn (x) and let Eˆ n be the completion of En with respect to pˆn . Denote by ρˆn and ρˆnm the extenˆ sions to the completions of ρn and ρnm respectively. The set Uˆ n := ρn (Un )En is the unit

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ball of Eˆ n . Finally, for every n, m ∈ N, n
m, set Rn : L(En , X) → L(E, X), f → f ◦ ρn and Rnm : L(En , X) → L(Em , X), f → f ◦ ρnm . Then (L(En , X), Rnm )n,m∈N,n
m is an injective inductive spectrum and L(E, X) is algebraically isomorphic to indn L(En , X). The isomorphism is constructed in the following way: if f ∈ L(E, X), there exist n ∈ N and a constant C > 0 such that f (x)
Cpn (x) for every x ∈ E. Define fn ∈ L(En , X) by fn (x + ker(pn )) := f (x). The map φ : L(E, X) → indn L(En , X), f → fn is an algebraical isomorphism. By definition, Lω (E, X) carries the inductive topology which is induced by indn Lb (En , X). This means that a fundamental family of 0-neighbourhoods is given by the sets 

   λn Rn W ρn (Un ), BX U := Γ , (λn )n∈N ∈ RN . n∈N

Let us consider now the case in which X is a Banach space. For every n, m ∈ N, n
m, set Rˆ n : L(Eˆ n , X) → L(E, X), f → f ◦ ρˆn and Rˆ nm : L(Eˆ n , X) → L(Eˆ m , X), f → f ◦ ρˆnm . Then (L(Eˆ n , X), Rˆ nm )n,m∈N,n
m is an injective inductive spectrum and L(E, X) is algebraically isomorphic to indn L(Eˆ n , X). The isomorphism is given in the following way: as before, if f ∈ L(E, X), then there exist n ∈ N and a constant C > 0 such that f (x)
Cpn (x) for every x ∈ E. Define fn ∈ L(En , X) by fn (x + ker(pn )) := f (x) and let fˆn be the extension to the completion of fn . The map φˆ : L(E, X) → indn L(Eˆ n , X), f → fˆn is an algebraical isomorphism. Hence we can topologize L(E, X) by the inductive topology τi which is induced by indn Lb (Eˆ n , X). This means that a fundamental family of 0-neighbourhoods is given by the sets 

   ˆ ˆ U := Γ λn Rn W Un , BX , (λn )n∈N ∈ RN . n∈N

The space (L(E, X), τi ) is denoted by Li (E, X). Clearly, if X is a Banach space, then Li (E, X) = Lω (E, X). Let F be a Fréchet space and let (Vh )h∈N be a decreasing fundamental sequence of 0-neighbourhoods in F , (qh )h∈N their Minkowski functionals. For every h ∈ N set Fh := F / ker(qh ), σh : F → Fh , x → x + ker(qh ) and σhk : Fk → Fh , x + ker(qk ) → x + ker(qh ), if k  h. Endow Fh with the norm qˆh (x + ker qn ) := qh (x) and let Fˆh be the completion of Fh with respect to this norm. Denote by ih the inclusion of Fh into Fˆh . Let σˆ h and σˆ hk ˆ be the extensions to the completions of σh and σhk respectively. The set Vˆn := σn (Vn )Fn is the unit ball of Fˆn . For every h, k ∈ N, h
k, set Sh : L(E, F ) → L(E, Fh ),   Sˆh : L(E, F ) → L E, Fˆh , Shk : L(E, Fk ) → L(E, Fh ),     Sˆhk : L E, Fˆk → L E, Fˆh ,

f → σh ◦ f, f → σˆ h ◦ f, f → σhk ◦ f f → σˆ hk ◦ f.

Then L1 = (L(E, Fh ), Shk )h,k∈N,h
k and L2 = (L(E, Fˆh ), Sˆhk )h,k∈N,h
k are projective spectra and L(E, F ) is algebraically isomorphic to Proj0 L1 and to Proj0 L2 (see [9, 39.8.10]). Moreover, Lω (E, F ) = projh Lω (E, Fh ). Define (L(E, F ), τi ) := Li (E, F ) := projh Li (E, Fˆh ).

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Finally for every n, k ∈ N, set Rnk : L(En , Fk ) → L(E, Fk ),     Rˆ nk : L Eˆ n , Fˆk → L E, Fˆk ,

f → f ◦ ρn , f → f ◦ ρˆn .

Remark 1. (1) The topology of Li (E, F ) does not depend on the sequences of 0-neighbourhoods that have been fixed in E and F . ˆ F ). (2) If Eˆ is the completion of E, then Li (E, F ) = Li (E, The spectrum L2 has the advantage of being a spectrum of (LB)-spaces, hence can be investigated by applying methods of functor theory. Some properties of the space Li (E, F ) are known and are collected in the following theorem. We recall the definition of the functor Ext1 . Let F be a Fréchet space. An injective resolution of F is an exact sequence i0

i1

0 → F → I0 → I1 → I2 → · · · ,

(1)

where Ik is an injective Fréchet space for all k. Every Fréchet space has an injective resolution. If E and F are Fréchet spaces, then, by definition, for any injective resolution (1) Ext1 (E, F ) = ker j1 / im j0 , where jk : L(E, Ik ) → L(E, Ik+1 ) is defined by jk (A) = ik ◦ A for A ∈ L(E, Ik ) (k = 1, 2). We refer to [18,19] for further information about the functor Ext1 . Theorem 2 [6,10,18]. Let E = projn (En , ·n ) and F = projn (Fn , |·|n ) be reduced projective limits of Banach spaces, E a proper Fréchet space. Consider the following assertions: (i) (ii) (iii) (iv) (v) (vi)

Ext1 (E, F ) = 0; each exact sequence 0 → F → G → E → 0 splits; Proj1 (L2 ) = 0; Li (E, F ) is ultrabornological; Li (E, F ) is barrelled; the pair (E, F) satisfies the condition (S2 )∗ : ∀n ∃N, m ∀M, l ∃S > 0, L ∀x ∈ En , y ∈ Fm :   xM |y|∗m
S xL |y|∗l + xN |y|∗n .

Then (1) (i) ⇔ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v), (i) ⇒ (vi); if Lb (E, F ) = Li (E, F ), then (v) ⇒ (vi); (2) if E or F is nuclear, or E = λ1 (A) or F = λ∞ (A), or both are Köthe echelon spaces, then (vi) implies (i). The results in (1) were proved by Vogt in [18, Theorem 1.8, Theorem 3.9(1)], in [17] (see also [19, Theorem 3.3.4, Proposition 5.1.5]) and [16, Proposition 4.4]. The equivalence of (v) and (i) in the cases described in (2) was proved in [6, Theorem 3.1] and in [10].

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Proposition 3. Let E be a metrizable locally convex space and F be a Fréchet space. (1) The map Φ : Lω (E, F ) → Li (E, F ),

(fh )h∈N → (ih ◦ fh )h∈N ,

is a continuous bijective map, such that Φ −1 is locally bounded. (2) In the following cases, Φ is a topological isomorphism: (a) F is a quojection, (b) the equality Lω (E, F ) = Lb (E, F ) holds topologically, (c) Li (E, F ) is bornological. Proof. (1) It is easily proved that Φ is linear and injective. Let (gh )h∈N ∈ Li (E, F ). Then, for every x ∈ E, (gh (x))h∈N ∈ projh Fˆh = projh Fh , hence, gh (x) ∈ Fh for every h ∈ N. For each h ∈ N, set fh := gh . Then (fh )h∈N ∈ Lω (E, F ) and Φ((fh )h∈N ) = (gh )h∈N . Therefore Φ is surjective. Let U be a 0-neighbourhood in Li (E, F ). Without loss of generality we can assume that there exist k ∈ N and (λn ) ∈ RN such that 


    k ˆ ˆ ˆ λn Rn W Un , Vk U = (gh )h∈N ∈ Li (E, F ): gk ∈ Γ . n∈N

Set


V = (fh )h∈N ∈ Lω (E, F ): fk ∈ Γ



λn Rnk

    W ρn (Un ), Vk .

n∈N

Then V is a 0-neighbourhood in Lω (E, F ) and Φ(V) ⊆ U . Therefore Φ is continuous. Li (E, F ). Then there exists a sequence (Bk )k∈N ∈  Let B be a bounded subset of  ˆk )) such that B ⊆ B(L (E, F i k∈N k Bk . By [15, 1.2] for every k ∈ N there exist λk > 0 k k (W (Uˆ n(k) , Vˆk )). Set Ck := λk Rn(k) (W (ρn(k) (Un(k) ), and n(k) ∈ N such that Bk ⊆ λk Rˆ n(k) Vˆk )). If (gk )k∈N ∈ B, then, for every k ∈ N, there exists φk ∈ L(Eˆ n(k) , Fˆk ) such that k gk = Rˆ n(k) (φk ) = φk ◦ ρˆn(k) and φk (λk ρˆn(k) (Un(k) )) ⊆ Vˆk . As before, for every x ∈ E set (fk (x))k := ϕ((gk (x))k ; then Φ((fh )h∈N ) = (gh )h∈N . For every k ∈ N, consider ψk := φk |En(k) . If y = ρn(k) (x) ∈ En(k) , then ψk (y) = φk ◦ ρn(k) (x) = φn(k) ◦ ρˆn(k) (x) = gk (x) ∈ ˆ Fk , hence ψk ∈ L(En(k)  , Fk ), fk = ψk ◦ ρn(k) and ψk (λk ρn(k) (Un(k) )) ⊆ Vk ∩ Fk = σk (Vk ). Therefore (fh )h∈N ∈ h Ch ∈ B(Lω (E, F )). (2)(a) If F is a quojection, then there exists a fundamental sequence of seminorms such that Fn is complete, hence clearly Lω (E, F ) = Li (E, F ). (2)(b) The assertion follows by observing that the inclusions Lω (E, F ) → Li (E, F ) → Lb (E, F ) are continuous. (2)(c) If Li (E, F ) is bornological, then Φ −1 is continuous, since it is locally bounded. 2 Remark 4. Analogously to the proof of (1), one shows that Lω (E, F ), Li (E, F ) and Lb (E, F ) have the same bounded sets.

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Example 5. The condition in (2)(b) is satisfied, e.g., when there exists a fundamental sequence of 0-neighbourhoods in F such that the space Lb (E, Fh ) is bornological (according to the previous notation). Indeed in this case Lb (E, F ) = projh Lb (E, Fh ) = projh Lω (E, Fh ) = Lω (E, F ). This holds when E is a (QNo) space (see [13, Corollary 4.3]) or a distinguished echelon space of order 1 (see [1, Corollary 7]).

3. Consequences for spaces of N -homogeneous polynomials By considering N -homogeneous polynomials instead of holomorphic mappings in [5, Proposition 3.27] and by observing that τω = τδ on P(N E, F ), we obtain the proof the following proposition. Proposition 6. Let E, F be complex locally convex spaces, N ∈ N. Then (P(N E, F ), τω ) is topologically isomorphic to Lω (⊗N,s,π E, F ). Analogously to the topology τi on spaces of linear continuous functions, if E and F are complex locally convex spaces, F complete, we define for every N ∈ N        N ˆ N,s,π E, F P E, F , τˆω := projα∈cs(F ) Pω E, Fˆα = Li ⊗ ˆ with where for every α ∈ cs(F ), Fˆα is the completion of the normed space (F / ker α, α), α(x ˆ + ker α) = α(x). Proposition 7. Let E and F be complex Fréchet spaces and let N ∈ N. If (P(N E, F ), τˆω ) is bornological, then (P(N E, F ), τω ) = (P(N E, F ), τˆω ). Proof. Indeed, if (P(N E, F ), τˆω ) is bornological, then by Proposition 3(2)(c),   N        ˆ N,s,π E, F = Lω (⊗N,s,π E, F ) = P N E, F , τω . P E, F , τˆω = Li ⊗

2

Proposition 8. Let E and F be complex Fréchet spaces, N ∈ N. Consider the following assertions: (i) (ii) (iii) (iv) (v) (vi)

ˆ N,π E, F ) = 0, Ext1 (⊗ ˆ N,s,π E, F ) = 0, Ext1 (⊗ (P(N E, F ), τˆω ) is ultrabornological, (P(N E, F ), τω ) is ultrabornological, (P(N E, F ), τω ) is barrelled, ˆ N,s,π E, F ) satisfies (S2 )∗ . the pair (⊗

Then (1) (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) and (ii) ⇒ (vi); (2) if E is stable (i.e., E × E is topologically isomorphic to E), then (i) ⇔ (ii); (3) if E or F is nuclear or E is a stable Köthe echelon space of order 1 or F is a Köthe echelon space of order ∞, then (i) ⇒ (ii) ⇔ (vi).

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ˆ N,s,π E, F ) is ultraProof. (1)(ii) ⇒ (iii). If (ii) holds, then (P(N E, F ), τˆω ) = Li (⊗ bornological by Theorem 2. The implication (iii) ⇒ (iv) follows from Proposition 7, while (iv) ⇒ (v) holds in general. Finally the implication (ii) ⇒ (vi) follows from Theorem 2. ˆ N,π E when E is stable (see [4, Theorem 3]). ˆ N,s,π E = ⊗ (2) holds, since ⊗ ˆ N,π E, F ) satisfies (S2∗ ), and therefore the pair (3) If (i) holds, then the pair (⊗ ∗ ˆ ˆ N,π E. Hence (i) imˆ (⊗N,s,π E, F ) satisfies (S2 ), since ⊗N,s,π E is a subspace of ⊗ plies (vi). We prove that (vi) implies (ii). If F is nuclear or a Köthe echelon space of order ∞, (vi) ⇒ (ii) by Theorem 2(2). ˆ N,s,π E is nuclear by [7, 21.2.3]. If A = (an )n∈N is a Köthe If E is nuclear, then ⊗ 1 ˆ matrix, then ⊗N,π λ (A) is isomorphic to λ1 (⊗N A), where ⊗N A = (anN )n∈N . If λ1 (A) is ˆ N,s,π λ1 (A) is isomorphic to λ1 (⊗N A), thus it is a Köthe echelon space stable, then also ⊗ of order 1. In both cases, (vi) ⇒ (ii) by Theorem 2(2). 2 Remark 9. Observe that in general if Ext1 (E, F ) = 0, one cannot expect that ˆ n,π E, F ) = 0. Indeed, in [11] it is shown that there exist Fréchet spaces E, F Ext1 (⊗ ˆ π G, F ) = 0. and G such that Ext1 (E, F ) = 0, Ext1 (G, F ) = 0 but Ext1 (E ⊗

4. Examples We recall the definition of some topological invariants (see, e.g., [12,16]). A Fréchet space E with a fundamental sequence of seminorms ( · n )n∈N is said to satisfy the property (DN) if 1 xm
Crxn0 + xn ; r the space E is said to verify the property (Ω) if ∃n0 ∀m ∃n, C > 0 ∀r > 0 ∀x ∈ E :

∀p ∃q ∀k ∃C > 0 ∀r > 0 ∀y ∈ E  :

1 y∗q
y∗p + Cry∗k ; r

finally, E has property (Ω) if the following holds: ∀p ∀ε > 0 ∃q ∀k ∃C > 0 ∀y ∈ E  :

y∗q 1+ε
Cy∗p ε y∗k .

When E is a Fréchet nuclear space, then E satisfies (DN) (respectively (Ω)) if and only if it is a subspace (respectively a quotient) of the space of rapidly decreasing sequences s (see [14]). Let us recall that a Fréchet space of type 2 is a projective limit of Banach spaces of type 2 (see [2]). We recall that Hilbert spaces and Lp (µ) with 2
p < ∞ are Banach spaces of type 2, hence, in particular, nuclear Fréchet spaces are of type 2. Lemma 10. Let E be a Fréchet space of type 2 or a Köthe echelon space. Then E satisfies ˆ N,π E satisfies (DN). the property (DN) if and only if ⊗ ˆ N,π E. The Proof. The “only if” part holds in both cases, since E is complemented in ⊗ “if” part follows by [11, Propositions 2.1,2.3]. 2

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Proposition 11. (1) Let E be a proper complex Fréchet space of type 2 or a Köthe echelon space and let F be a proper complex Fréchet space. If E satisfies (DN) and F satisfies (Ω) and (a) F is a nuclear Fréchet space or a Köthe space λ∞ (A), or (b) E is a nuclear Fréchet space or a Köthe echelon space λ1 (A), then (P(N E, F ), τω ) is ultrabornological for every N ∈ N. (2) Let E and F be nuclear spaces. (a) if F = Λr (α) (r = 1, ∞), supn∈N αn+1 /αn < +∞ and E is stable, then the following conditions are equivalent: (i) (P(N E, F ), τω ) is ultrabornological, (ii) (P(N E, F ), τω ) is barrelled, (iii) E satisfies the property (DN); (b) if E = Λ∞ (α), supn∈N α2n /αn < +∞, then the following conditions are equivalent: (i) (P(N E, F ), τω ) is ultrabornological, (ii) (P(N E, F ), τω ) is barrelled, (iii) F satisfies the property (Ω); (c) if E = Λ1 (α), supn∈N αn+1 /αn = 1, then the following conditions are equivalent: (i) (P(N E, F ), τω ) is ultrabornological, (ii) (P(N E, F ), τω ) is barrelled, (iii) F satisfies the property (Ω). ˆ N,π,s E satisfy the property (DN). The ˆ N,π E and hence ⊗ Proof. (1) By Lemma 10, ⊗ assertions follow by [18, Theorem 5.1] and by Proposition 8. (2) Note that, since E is nuclear, by the considerations in Example 5 and by Proposiˆ N,s,π E, F ). tion 6, (P(N E, F ), τω ) is topologically isomorphic to Lb (⊗ (a) By the previous remarks, Proposition 8 and by [16, Theorem 4.9(3)α], the space ˆ N,π E = ⊗ ˆ N,s,π E satisfies (DN), hence if (P(N E, F ), τω ) is bornological if and only if ⊗ and only if E verifies (DN), by Lemma 10. (b) Observe that Λ∞ (α) is stable, by [12, 29 Aufgaben 3(a)] and that ˆ N,π Λ∞ (α) = Λ∞ (Nα). ⊗ Moreover supn∈N (Nαn+1 )/(Nαn )
supn∈N α2n /αn < +∞. The assertion follows by [16, Theorem 4.9(3)γ ] (c) Λ1 (α) is stable, because sup α2n /αn
sup n∈N

n 

αn+k /αn+k−1
1.

n∈N k=1

The claim follows as in (b). 2 Remark 12. Observe that the conditions in (2)(a) and (2)(b) in the previous proposition are exactly those that ensure that the pair (F, E) has the localization property, i.e., every linear continuous map from F into E is bounded (see [3,15]). But in general, if (P(N E, F ), τω )

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is ultrabornological, the pair (F, E) may not have the localization property. For example, consider E = F = s. On the other hand, if E is nuclear and (F, E) has the localization ˆ π F is barrelled, by [15, Theorem 7.3] (see property, then Li (E, F ) = Lb (E, F ) = Eb ⊗ also [3, Theorem 6]). Other remarks on the connections between the localization property and the barrelledness of spaces of linear continuous functions between Fréchet spaces can be found in [16], remarks after Theorem 4.7 and in [3].

5. Consequences for spaces of holomorphic functions In this section we obtain some information on spaces of holomorphic functions between Fréchet spaces by combining the results of the previous section with results about the decomposition of holomorphic functions. We recall that a sequence of subspaces {En }n of a locally convex space E is an S-absolute decomposition of E if (a) for each x ∈ Ethere exists a unique sequence of vectors (xn )n , xn ∈ En for all n ∈ N, such that x = ∞ n=1 xn ;  m (b) the projections (un )n defined by um ( ∞ n=1 xn ) := n=1 xn are continuous; 1/n
1 and for all x = (c) for all scalar sequence (α ) such that lim sup n n n→∞ |αn | ∞ ∞ n=1 xn ∈ E, xn ∈ En for all n ∈ N, n=1 αn xn ∈ E; (d) for all scalar sequence (αn )n such that lim supn→∞ |αn |1/n
1 and for all p ∈ cs(E), 

∞ ∞ xn := |αn |p(xn ) pα n=1

n=1

defines a continuous seminorm on E. Lemma 13. Let E = projn (En , Rn,n+1 ) be a projective limit of locally convex spaces and for every n ∈ N let {Enk }k∈N be an S-absolute decomposition for En such that k ) ⊆ E k for every k, n ∈ N. Then (proj E k ) Rn,n+1 (En+1 n n k∈N is an S-absolute decompon sition of E. a unique (xnk )k , xnk ∈ Enk Proof. Let x =  (xn )n ∈ E. For every n ∈ N, there ∞exists ∞ sequence ∞ k k k such that xn = k=1 xn in En , hence x = ( k=1 xn )n = k=1 (xn )n in E. Moreover, (xnk )n ∈ projn Enk for every k ∈ N. Analogously, one can show that the projections are continuous and that conditions (c) and (d) of the definition are satisfied. 2 Let E and F be complex Fréchet spaces and U a balanced open subset of E. For τ = τω , τδ , consider the projective spectra      Hτ = H U, Fˆα , τ , φα,β α,β∈cs(F ),α
β , where Fˆα is the completion of the normed space (F / ker α, α), ˆ with α(x ˆ + ker α) = α(x) and, for every α, β ∈ cs(F ), α
β, the map φα,β : H(U, Fˆβ ) → H(U, Fˆα ) is defined by φα,β (f ) = f ◦ σα,β where σα,β : Fˆβ → Fˆα is the canonical map. Then H(U, F ) is

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algebraically isomorphic to Proj0 (Hτ ). (This can be seen, e.g., by applying [5, Proposition 3.27] and [9, 39.8.10]) Set       (τ = τω , τδ ). H(U, F ), τˆ = projα∈cs(F ) H U, Fˆα , τ Clearly τˆ is coarser than τ and they coincide for example when F is a quojection. Moreover τˆω
τˆδ and they induce the same topology τˆω on the spaces P(N E, F ). Hence, by the previous considerations on spaces of polynomials, in general it may happen that τˆ is strictly coarser than τ . Proposition 14. Let E and F be complex Fréchet spaces and U a balanced open subset of E. Then (1) {(P(N E, F ), τˆω )}N∈N is an S-absolute decomposition for (H(U, F ), τˆω ) and for (H(U, F ), τˆδ ); (2) for every N ∈ N, the space (P(N E, F ), τˆω ) is a closed complemented subspace of (H(U, F ), τˆω ) and of (H(U, F ), τˆδ ). Proof. The assertion (1) follows by the previous lemma and by [5, Proposition 3.36]. The assertion (2) is an easy consequence of [5, Proposition 3.22]. 2 Proposition 15. Let E and F be complex Fréchet spaces and let U be a balanced open subset of E. Then (1) if (H(U, F ), τˆω ) or (H(U, F ), τˆδ ) are bornological (respectively ultrabornological) then (P(N E, F ), τˆω ) and (P(N E, F ), τˆω ) are bornological (respectively ultrabornological) for every N ∈ N; (2) if for every N ∈ N, (P(N E, F ), τˆω ) is bornological, then {(P(N E, F ), τˆω )}N∈N is an S-absolute decomposition for (H(U, F ), τˆωbor ) and for (H(U, F ), τˆδbor ); in this case, if (fβ )β∈Γ is a τˆδ -bounded net in H(U, F ), then the following are equivalent: (i) fβ → 0 as β → ∞ in (H(U, F ), τˆδbor ); (ii) fβ → 0 as β → ∞ in (H(U, F ), τˆδ ); (iii) fβ → 0 as β → ∞ in (H(U, F ), τˆωbor ); (iv) fβ → 0 as β → ∞ in (H(U, F ), τˆω ); (v) dˆ N fβ (0)/N! → 0 as β → ∞ in (P(N E, F ), τˆω ) for every N ∈ N. Proof. (1) follows by Proposition 14(1). (2) follows by [5, Proposition 3.34], by observing that τˆω
τˆδ , τˆω
τˆωbor
τˆδbor and that τˆδbor and τˆδ have the same bounded sets. 2 Acknowledgments The author thanks the anonymous referee for several helpful remarks that improved the original version of the paper. The author also thanks Professor J. Bonet for his suggestions.

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