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)     

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 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.     

Bases and quasi‐reflexivity in Fréchet spaces

A Fréchet space E is quasi‐reflexive if, either dim(E″/E) < ∞, or E″[β(E″,E′)]/E is isomorphic to ω. A Fréchet space E is totally quasi‐reflexive if every separated quotient is quasi‐reflexive. In this paper we show, using Schauder bases, that E is totally quasi‐reflexive if and only if it is isomorphic to a closed subspace of a countable product of quasi‐reflexive Banach spaces. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Convolution operators on spaces of entire functions

We show that nontrivial convolution operators on certain spaces of entire functions on E are frequently hypercyclic when E is a normed space and when E is the strong dual of a Fréchet nuclear space. We also obtain results of existence and approximation for convolution equations on certain spaces of entire functions on arbitrary locally convex spaces.     

Holomorphic functions on strict inductive limits

We study strict inductive limits of Fréchet Montel (FM) spaces and reflexive Fréchet (RF) spaces and we obtain some interesting examples in the theory of infinite dimensional holomorphy. P_M(^kE′) and P_HY(^kE′) will denote respectively the set of all k-homogeneous polynomials on E′ that are bounded on bounded sets and the set of all k-homogeneous polynomials on E′ that are continuous on compact sets. ?_SM(^kE′) is the space of all symetric k -multilinear mappings from E′ × ... × E′ into C that are bounded on bounded sets. H_HY(E′) will denote the set of all G-analytic functions on E′ that are continuous on the compact subsets of E′.     

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.     

Counterexample to Peano's theorem in infinite-dimensionalF′-spaces

LetE be a nonnormable Fréchet space, and letE′ be the space of all continuous linear functionals onE in the strong topology. A continuous mappingf:E′→E′ such that for anyt ₀∈ℝ,x ₀∈E′, the Cauchy problemx=f(x), x(t ₀ )=x ₀ has no solutions is constructed. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 128–137, July, 1997. Translated by V. N. Dubrovsky     

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.     

Polynomial reflexivity in Banach spaces

We ask when the space ofN-homogeneous analytic polynomials on a Banach space is reflexive. This turns out to be related to whether polynomials are weakly sequentially continuous, and to the geometry of spreading models. For example, if these spaces are reflexive for allN, no quotient of the dual space may have a spreading model with an upperq-estimate, and every bounded holomorphic function on the unit ball has a Taylor series made up of weakly sequentially continuous polynomials (we assume the approximation property). Alencar, Aron and Dineen [AAD] gave the first example of some properties of a polynomially reflexive space (usingT*, the original Tsirelson space); we show that these properties and others are shared by all polynomially reflexive spaces. This paper forms a portion of the Ph. D. dissertation of the author, under the supervision of W. B. Johnson.     

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.     

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     

Limitations to Fréchet’s metric embedding method

Fréchet’s classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Fréchet embedding is Bourgain's embedding [4]. The authors have recently shown [2] that for every ε>0, anyn-point metric space contains a subset of size at leastn ^1−ε which embeds into ℓ₂ with distortion . The embedding used in [2] is non-Fréchet, and the purpose of this note is to show that this is not coincidental. Specifically, for every ε>0, we construct arbitrarily largen-point metric spaces, such that the distortion of any Fréchet embedding into ℓ_p on subsets of size at leastn ^1/2+ε is Ω((logn)^1/p ). Supported in part by a grant from the Israeli National Science Foundation. Supported in part by a grant from the Israeli National Science Foundation. Supported in part by the Landau Center.     

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     

Weakly continuous mappings on Banach spaces

It is shown that every n-homogeneous continuous polynomial on a Banach space E which is weakly continuous on the unit ball of E is weakly uniformly continuous on the unit ball of E. Applications of the result to spaces of polynomials and holomorphic mappings on E are given.     

Quasi-reflexive Fréchet spaces and contractively power bounded operators

In this paper we introduce a new class of operators acting on a locally convex space. We show that for some Fréchet spaces all these operators are mean ergodic. This leads to the conclusion that the classes of reflexive and non-reflexive Fréchet spaces are, in a sense, close to each other.     

Intersections of Fréchet Schwartz spaces and their duals

In this article the structure of the intersections of a Fréchet Schwartz space F and a (DFS)-space E=ind _n E _n is investigated. A complete characterization of the locally convex properties of E ⋃ F is given. This space is boraological if and only if the inductive limit E + F is complete. The results are based on recent progress on the structure of (LF)-spaces. The article includes examples of (FS)-spaces F and (DFS)-spaces E such that there are sequentially continuous linear forms on E ⋃ F which are not continuous, thus answering a question of Langenbruch. Acknowledgement: The results in this article were obtained during the author’s stay at the University of Paderborn, Germany, during the academic year 1994/95. The support of the Alexander von Humboldt Stiftung is greatly appreciated. The content of the article was presented as an invited paper in a Special Session of the AMS meeting in New York in April, 1996.     

A sufficient condition of type (Ω) for tame splitting of short exact sequences of Fréchet spaces

In a previous paper, the quotient spaces of (s) in the tame category of nuclear Fréchet spaces have been characterized by property (ΩDZ) corresponding to the topological condition (Ω) of D. Vogt and M. J. Wagner. In addition, a splitting theorem has been proved which provides the existence of a tame linear right inverse of a tame linear map on the assumption that the kernel of the given map has property (ΩDZ) and that certain tameness conditions hold. In this paper it is proved that property (Ω) in standard form (i.e., the dual norms ‖ ‖ _n ^* are logarithmically convex) implies the tame splitting condition (ΩDZ) for any tamely nuclear Fréchet space equipped with a grading defined by sermiscalar products. As an application, property (ΩDZ) is verified for the kernels of any hypoelliptic system of linear partial differential operators with constant coefficients on ℝ^N or on a bounded convex region in ℝ^N.     

An operator on a Fréchet space with no common invariant subspace with its inverse

A Fréchet space with a two-sided Schauder basis is constructed, such that the corresponding bilateral shift is continuous and invertible, and has no common nontrivial invariant subspace with its inverse. This shows in particular, that the problem of existence of hyperinvariant subspaces for operators on general Fréchet spaces, admits a negative answer. It is also shown that the dual of the Fréchet space constructed can be identified with a commutative locally convex complete topological algebra with unit, which has no closed nontrivial ideals.