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 mackey convergence in locally convex spaces

After some introductory propositions, we give a dual characterization of those locally convex spaces which satisfy the Mackey convergence condition or the fast convergence condition by means of Schwartz topologies. Making use of the universal Schwartz space (l _∞ ,τ(l _∞ ,l ₁)) we prove some representation theorems for bornological and ultrabornological spaces, that is, every bornological spaceE is a dense subspace of an inductive limit lim indE _a, a∈A, ofseparable Banach spacesE _a, and every Mackey null sequence inE is a null sequence in someE _a. IfE is ultrabornological, thenE can be represented as lim indE _a,a∈A, allE _a separable Banach spaces, such that every fast null sequence inE is a null sequence in someE _a.     

Weakly K-analytic spaces and the three-space property for analyticity

Let (E,E^′) be a dual pair of vector spaces. The paper studies general conditions which allow to lift analyticity (or K-analyticity) from the weak topology σ(E,E^′) to stronger ones in the frame of (E,E^′). First we show that the Mackey dual of a space Cp(X) is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of Cp(X) is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space NN. This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. Several questions, remarks and examples are included.     

Thin and thick sets in locally convex spaces

We introduce in this work some normed space notions such as norming, thin and thick sets in general locally convex spaces. We also study some effects of thick sets on the uniform boundedness-like principles in locally convex spaces such as “weak*-bounded sets are strong*-bounded if and only if the space is a Banach–Mackey space”. It is proved that these principles occur under some weaker conditions by means of thick sets. Further, we show that the thickness is a duality invariant, that is, all compatible topologies for some locally convex space have the same thick sets.     

On countable strict inductive limits

In this paper we prove that if E is the strict inductive limit of a sequence of Mackey spaces {E_n} such that for every positive integer n, the topological dual space of E_n, E′_n, provided with the Mackey topology μ(E′_n,E_n), is ultrabornological, then the topological dual space E′ of E, provided with the Mackey topology μ(E′,E), is ultrabornological. We also prove that if E is a strict (LF)-space and G a closed subspace of E′ [μ(E′,E)] such that E′[μ(E′,E)] /G is sequentially complete, then E′[μ(E′,E)]/G is complete.     

DENTABILITY IN LOCALLY CONVEX SPACES

Abstract

Let A be a non-empty bounded subset of a locally convex space E. We show that if all the separable subsets of A are weakly metrisable, then the weak*-compact subsets of E¹ satisfy geometrical conditions which are similar to the concept of “dentability” used to characterise the Radon-Nikodý Property in dual Banach spaces.     

DIEUDONNÉ—KÖTHE DUALITY FOR VECTOR—VALUED FUNCTION SPACES: LOCALIZATION OF BOUNDED SETS AND BARRELLEDNESS

Abstract

We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.     

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.     

Asplund operators and holomorphic maps

LetE andF be locally convex topological vector spaces. A holomorphic mapf: E→F is defined to be an Asplund map if it takes the separable subsets of a neighbourhood of eacha∈E into absolutely convex weakly metrisable subsets ofF; a Banach space is an Asplund space if and only if its identity map has this property. We show that a continuous linear map from a quasinormable locally convex spaceE into a Banach spaceF is an Asplund map if and only if it factors through an Asplund space. IfE andF are both Banach spaces, then a holomorphic mapf: E→F is an Asplund map if and only if its derivative maps factor through Asplund spaces for eacha∈E. This is true if and only if such a factorisation holds ata=0. Part of this research was done during a visit to the University of Namibia, whose financial support is gratefully acknowledged This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990     

Multi-hypercyclic operators are hypercyclic

Herrero conjectured in 1991 that every multi-hypercyclic (respectively, multi-supercyclic) operator on a Hilbert space is in fact hypercyclic (respectively, supercyclic). In this article we settle this conjecture in the affirmative even for continuous linear operators defined on arbitrary locally convex spaces. More precisely, we show that, if is a continuous linear operator on a locally convex space E such that there is a finite collection of orbits of T satisfying that each element in E can be arbitrarily approximated by a vector of one of these orbits, then there is a single orbit dense in E. We also prove the corresponding result for a weaker notion of approximation, called supercyclicity . Received October 18, 1999 / Published online February 5, 2001     

On tree characterizations of <Emphasis Type="Italic">G</Emphasis><Subscript>δ</Subscript>-embeddings and some Banach spaces

We show that a one-to-one bounded linear operator T from a separable Banach space E to a Banach space X is a G _δ-embedding if and only if every T-null tree in S _E has a branch which is a boundedly complete basic sequence. We then consider the notions of regulators and skipped blocking decompositions of Banach spaces and show, in a fairly general set up, that the existence of a regulator is equivalent to that of special skipped blocking decomposition. As applications, the following results are obtained. (a) A separable Banach space E has separable dual if and only if every w*-null tree in S _E * has a branch which is a boundedly complete basic sequence. (b) A Banach space E with separable dual has the point of continuity property if and only if every w-null tree in S _E has a branch which is a boundedly complete basic sequence. We also give examples to show that the tree hypothesis in both the cases above cannot be replaced in general with the assumption that every normalized w*-null (w-null in (b)) sequence has a subsequence which is a boundedly complete basic sequence. The research of S. Dutta was supported in part by the Institute for Advanced Studies in Mathematics at Ben-Gurion University of the Negev. The research of V. P. Fonf was supported in part by Israel Science Foundation, Grant No. 139/03.     

The Hahn-Banach Property and the Axiom of Choice

We work in set theory ZF without axiom of choice. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gateaux-differentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → ? is a linear functional such that f ≤ p|F then there exists a linear functional g : E → ? such that g extends f and g ≤ p. We also prove that the continuous Hahn-Banach property on a topological vector space E is equivalent to the classical geometrical forms of the Hahn-Banach theorem on E. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn-Banach property on Gateaux-differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn-Banach property, they do not satisfy the whole Hahn-Banach property in ZF+DC.     

Sur la Topologie m -Convexe d’une algèbre localement A -convexe

In a locallyA-convex algebra (E, τ) we consider the associatedm-convex topologym(τ). We show that the completion ofE with respect tom(τ) is always a locallyA-convex algebra contained in the complete locally convex space obtained from (E, τ). The topologym(τ) is also used to characterize locally boundedly multiplicatively convex algebras among locallyA-convex ones.

     

A characterization of distinguished Fréchet spaces

Bierstedt and Bonet proved in 1988 that if a metrizable locally convex space E satisfies the Heinrich's density condition, then every bounded set in the strong dual (E ′, β (E ′, E)) of E is metrizable; consequently E is distinguished, i.e. (E ′, β (E ′, E)) is quasibarrelled. However there are examples of distinguished Fréchet spaces whose strong dual contains nonmetrizable bounded sets. We prove that a metrizable locally convex space E is distinguished iff every bounded set in the strong dual (E ′, β (E ′, E)) has countable tightness, i.e. for every bounded set A in (E ′, β (E ′, E)) and every x in the closure of A there exists a countable subset B of A whose closure contains x. This extends also a classical result of Grothendieck. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spaces of Vector-Valued Integrable Functions and Localization of Bounded Subsets

We study the structure of bounded sets in the space L¹{E} of absolutely integrable Lusin-measurable functions with values in a locally convex space E. The main idea is to extend the notion of property (B) of Pietsch, defined within the context of vector-valued sequences, to spaces of vector-valued functions. We prove that this extension, that at first sight looks more restrictive, coincides with the original property (B) for quasicomplete spaces. Then we show that when dealing with a locally convex space, property (B) provides the link to prove the equivalence between Radon–Nikodym property (the existence of a density function for certain vector measures) and the integral representation of continuous linear operators T: L¹ → E, a fact well-known for Banach spaces. We also study the relationship between Radon–Nikodym property and the characterization of the dual of L¹{E} as the space L^∞{E′_b}.     

A note on embedding into product spaces

Using factorization properties of an operator ideal over a Banach space, it is shown how to embed a locally convex space from the corresponding Grothendieck space ideal into a suitable power of E, thus achieving a unified treatment of several embedding theorems involving certain classes of locally convex spaces.     

Shrinking and boundedly complete Schauder frames in Fréchet spaces

We study Schauder frames in Fréchet spaces and their duals, as well as perturbation results. We define shrinking and boundedly complete Schauder frames on a locally convex space, study the duality of these two concepts and their relation with the reflexivity of the space. We characterize when an unconditional Schauder frame is shrinking or boundedly complete in terms of properties of the space. Several examples of concrete Schauder frames in function spaces are also presented.     

Schwartz spaces and compact holomorphic mappings

LetE be a locally convex space. We investigate under which conditions onE it is true that every holomorphic mapping fromE intoc ₀ is compact. We show that Schwartzity ofE is a sufficient condition and also a necessary condition ifE is quasi-normable.     

On some relationships between biorthogonal systems,linear topologies and the weak‐AFPP

In this paper, we obtain new results for the weak‐AFPP in abstract spaces by exploiting biorthogonal systems techniques. Firstly, we investigate the strong‐AFPP on countably infinite dimensional Hausdorff locally convex spaces. Spaces of this class are shown to be sequentially complete iff they have the hereditary FPP for totally bounded, closed convex sets. This might open a research line for the analysis of weak‐AFPP in such frames. In connection, we provide a simple criterion for the containement of ?₁‐sequences in terms of strongly‐equicontinuous biorthogonal systems. We then establish a few results concerning the existence of Hausdorff finer vector topologies on abstract spaces having as prescribed condition the existence of such systems. The proofs are based on methods of Peck and Porta concerning building of finer vector topologies, and a classical construction of Singer which allows us to prove under rather natural conditions the existence of equicontinuous biorthogonal systems in metrizable locally convex spaces. These results are compatible with the failure of the weak‐AFPP. We also study the inverse problem by proving that every infinite dimensional vector space admits a (non‐locally convex) Hausdorff vector topology which is complete, non‐metrizable and is compatible with a bounded Hamel Schauder basis. It is shown further that such a topology has the ‐AFPP, where is the linear span of coefficient functionals associated to a Hamel basis. Finally, inspired by a result of Shapiro, we observe that if X is a non‐locally convex F‐space with an absolute basis, then the weak‐AFPP is equivalent to the fact that every bounded convex subset of X is compact.