On the Injective Tensor Product of Distinguished Frechet Spaces

An example of two distinguished Fréchet spaces E, F is given (even more, E is quasinormable and F is normable) such that their completed injective tensor product E?F is not distinguished. On the other hand, it is proved that for arbitrary reflexive Fréchet space E and arbitrary compact set K the space of E - valued continuous functions C(K, E) is distinguished and its strong dual is naturally isomorphic to ? where L¹(μ) = C(K)¹.     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous. (Received 24 March 1999; in final form 14 February 2000)     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous.     

On Symmetric Schauder Bases in Non-Archimedean Fréchet Spaces

 It is proved that any infinite-dimensional non-archimedean Fréchet space with a symmetric basis is isomorphic to c ₀ or ?^ℕ. A similar result is shown for homogeneous bases. It is also proved that any infinite-dimensional nuclear non-archimedean Fréchet space with a subsymmetric basis is isomorphic to ?^ℕ. In fact, much stronger results are obtained.     

On the bidual of quasinormable Fréchet algebras

It is known that the bidual of a quasinormable Fréchet space E with local Banach spaces $(E_n)_{n\in {\mathbb N}}$ is topologically isomorphic to the inverse limit of $\big (E_n^{\prime \prime }\big )_{n\in {\mathbb N}}$. With the aid of the Arens product and by homological means, we prove that the previous result is equally valid for quasinormable Fréchet m‐convex algebras. This allows showing that the bidual of a σ‐C*‐algebra equipped with the Arens product is a σ‐C*‐algebra and presenting a new direct proof of a result on acyclic spectra due to Palamodov.     

On quotients of non‐Archimedean Fréchet spaces

It is proved when a non‐Archimedean Fréchet space E of countable type has a quotient isomorphic to ??^?, c^?₀ or c₀ × ??^?. It is also shown when E has a non‐normable quotient with a continuous norm. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reflexive operators with domain in Köthe spaces

It is proved that if a K?the space λ₁(A) is distinguished and E is an arbitrary Fréchet space then every reflexive map T: λ₁(A)→E (i.e., T maps bounded sets into relatively weakly compact ones) factorizes through a reflexive Fréchet space. An analogous result is proved for Montel maps (i.e., which map bounded sets into relatively compact ones). The result is a consequence of the fact proved also in this paper that, for a distinguished λ₁(A) space, the spaces of reflexive maps R(λ₁(A), C(K)) and of Montel maps M(λ₁(A), C(K)) are the Mackey completions of the spaces of weakly compact and compact maps, respectively. Consequences for spaces of vector-valued (weakly) continuous functions are also obtained. Received: 24 November 1997 / Revised version: 14 May 1998     

On Symmetric Schauder Bases in Non-Archimedean Fréchet Spaces

 It is proved that any infinite-dimensional non-archimedean Fréchet space with a symmetric basis is isomorphic to c ₀ or ?^ℕ. A similar result is shown for homogeneous bases. It is also proved that any infinite-dimensional nuclear non-archimedean Fréchet space with a subsymmetric basis is isomorphic to ?^ℕ. In fact, much stronger results are obtained. Received August 27, 2001; in revised form February 8, 2002     

The bounded approximation property for spaces of holomorphic mappings on infinite dimensional spaces

We study the bounded approximation property for spaces of holomorphic functions. We show that if U is a balanced open subset of a Fréchet–Schwartz space or (DFM )‐space E , then the space ??(U ) of holomorphic mappings on U , with the compact‐open topology, has the bounded approximation property if and only if E has the bounded approximation property. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two problems of Valdivia on distinguished Frechet spaces

In this paper we construct the following spaces: (a) a Fréchet spaceE with basis which is non-distinguished and has no subspace isomorphic to ℓ₁ (b) a non-distinguished Fréchet spaceF such that every separable subspace ofF is distinguished (even more, every separable subspace ofF has a separable dual). These examples answer in the negative two questions posed by Valdivia.     

Basic Sequences in the Dual of a Fréchet Space

In this paper we study some properties of basic sequences in the dual of a Fréechet space. As a consequence we obtain that if E is a Fréechet space with the property that for each closed subspace F of E and each bounded subset B of E/F there is a bounded subset A of E with φ(A) = B, where φ denotes the canonical surjection of E onto E/F, then one of the following conditions is at least satisfied: 1. E is a Banach space, 2. E is a Schwartz space, 3. E is the product of a Banach space by ω. Finally, we also obtain some results concerning totally reflexive spaces.     

An example of a nuclear space in infinite dimensional holomorphy

LetU be an open subset of a complex locally convex spaceE, andH(U) the space of holomorphic functions fromU toC. If the dualE′ ofE is nuclear with respect to the topology generated by the absolutely convex compact subsets ofE, then it is shown thatH(U) endowed with the compact open topology is a nuclear space. In particular, ifE is the strong dual of a Fréchet nuclear space, thenH(U) is a Fréchet nuclear space.     

Universal K?the Sequence Spaces

 A characterization is given for the K?the matrices B such that the K?the sequence space , with , contains all K?the sequence spaces of order p as subspaces. It follows that the class of K?the sequence spaces of order p has a universal element which is quasinormable. In particular, there is a quasinormable space (respectively, which contains every nuclear Fréchet space with basis (respectively, every countably normed Fréchet Schwartz space). Only Fréchet spaces with continuous norm are considered in this note.     

On closed subspaces with Schauder bases in non-archimedean Fréchet spaces

The main purpose of this paper is to prove that a non-archimedean Fréchet space of countable type is normable (respectively nuclear; reflexive; a Montel space) if and only if any its closed subspace with a Schauder basis is normable (respectively nuclear; reflexive; a Montel space). It is also shown that any Schauder basis in a non-normable non-archimedean Fréchet space has a block basic sequence whose closed linear span is nuclear. It follows that any non-normable non-archimedean Fréchet space contains an infinite-dimensional nuclear closed subspace with a Schauder basis. Moreover, it is proved that a non-archimedean Fréchet space E with a Schauder basis contains an infinite-dimensional complemented nuclear closed subspace with a Schauder basis if and only if any Schauder basis in E has a subsequence whose closed linear span is nuclear.     

Universal Köthe Sequence Spaces

 A characterization is given for the K?the matrices B such that the K?the sequence space , with , contains all K?the sequence spaces of order p as subspaces. It follows that the class of K?the sequence spaces of order p has a universal element which is quasinormable. In particular, there is a quasinormable space (respectively, which contains every nuclear Fréchet space with basis (respectively, every countably normed Fréchet Schwartz space). Only Fréchet spaces with continuous norm are considered in this note. Received 15 January 1997; in final form 9 June 1997     

Non-quasianalytic curves in Fr��chet spaces

It is proved that every weakly non-quasianalytic ultradifferentiable curve with values in a Fréchet space E is topologically (or strongly) ultradifferentiable if and only if the space E satisfies the topological invariant (DN), thus solving a problem posed by Kriegl and Michor.     

Some Uniqueness Results for Functional Damped Semilinear Differential Equations in Frechet Spaces

A recent nonlinear alternative for contraction maps in Frechet spaces due to Frigon and Granas （Resultats de type Leray-Schauder pour des contractions sur des espaces de Frechet, Ann. Sci. Math. Quebec 22, （2）, 161-168 （1998））, combined with semigroup theory, is used to investigate the existence and uniqueness of mild solutions for first- and second-order functional semi linear and neutral damped differential equations in Frechet space.     

Universal connections in Fréchet principal bundles

A new methodology leading to the construction of a universal connection for Fréchet principal bundles is proposed in this paper. The classical theory, applied successfully so far for finite dimensional and Banach modelled bundles, collapses within the framework of Fréchet manifolds. However, based on the replacement of the space of continuous linear mappings by an appropriate topological vector space, we endow the bundle J ¹ P of 1-jets of the sections of a Fréchet principal bundle P with a connection form by means of which we may “reproduce” every connection of P.      

Holomorphic liftings and Bergman kernel estimates for 𝒟ℱ𝒩‐domains

Let E be a 𝒟ℱ𝒩‐space and let U ⊂ E be open. By applying the nuclearity of the Fréchet space ℋ︁(U) of holomorphic functions on U we show that there are finite measures μ on U leading to Bergman spaces of μ ‐square integrable holomorphic functions. We give an explicit construction for μ by using infinite dimensional Gaussian measures. Moreover, we prove boundary estimates for the corresponding Bergman kernels K_μ on the diagonal and we give an application of our results to liftings of μ ‐square integrable Banach space valued holomorphic functions over U. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Web‐compact spaces,Fréchet‐Urysohn groups and a Suslin closed graph theorem

A topological space X is strongly web‐compact if X admits a family {A_α: α ∈ ?^?} of relatively countably compact sets covering X and such that A_α ? A_β for α ≤ β. The main result of this paper states the following: Theorem A Let X and Y be topological groups and f a homomorphism between X and Y with closed graph. If X is Fréchet‐Urysohn and Baire and Y is strongly web‐compact, then f is continuous. This extends a result of Valdivia. We provide an example showing that the property of being strongly web‐compact is not productive. This applies to show that there are quasi‐Suslin spaces X whose product X × X is not quasi‐Suslin (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)