Analysis of a problem of Raikov with applications to barreled and bornological spaces

Raikov’s conjecture states that semi-abelian categories are quasi-abelian. A first counterexample is contained in a paper of Bonet and Dierolf who considered the category of bornological locally convex spaces. We prove that every semi-abelian category I admits a left essential embedding into a quasi-abelian category K_l(I) such that I can be recovered from K_l(I) by localization. Conversely, it is shown that left essential full subcategories I of a quasi-abelian category are semi-abelian, and a criterion for I to be quasi-abelian is given. Applied to categories of locally convex spaces, the criterion shows that barreled or bornological spaces are natural counterexamples to Raikov’s conjecture. Using a dual argument, the criterion leads to a simplification of Bonet and Dierolf’s example.     

An Isometric Representation of the Dual of {\user1{\mathcal{C}}}{\left( {X,\mathbb{R}} \right)}

Many structures in functional analysis are introduced as the limit of an inverse (aka projective) system of seminormed spaces [2, 3, 8]. In these situations, the dual is moreover equipped with a seminorm. Although the topology of the inverse limit is seldom metrizable, there is always a natural overlying locally convex approach structure. We provide a method for computing the adjoint of this space, by showing that the dual of a limit of locally convex approach spaces becomes a co-limit in the category of seminormed spaces. As an application we obtain an isometric representation of the dual space of real valued continuous functions on a locally compact Hausdorff space X, equipped with the compact open structure.     

Grothendieck space ideals and weak continuity of polynomials on locally convex spaces

We introduce the classes of locally convex spaces with the local Dunford-Pettis property and locally dual Schur spaces. We examine their properties and their relationship to other classes of locally convex spaces. In the class of locally convex spaces with the local Dunford-Pettis property all polynomials are weakly sequentially continuous whereas in the class of locally dual Schur spaces all polynomials are weakly continuous on bounded sets. Research supported by Science Foundation Ireland, Basic Research Grant 2004.     

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.     

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.     

Fundamental results for pointfree convex geometry

Inspired by locale theory, we propose “pointfree convex geometry”. We introduce the notion of convexity algebra as a pointfree convexity space. There are two notions of a point for convexity algebra: one is a chain-prime meet-complete filter and the other is a maximal meet-complete filter. In this paper we show the following: (1) the former notion of a point induces a dual equivalence between the category of “spatial” convexity algebras and the category of “sober” convexity spaces as well as a dual adjunction between the category of convexity algebras and the category of convexity spaces; (2) the latter notion of point induces a dual equivalence between the category of “m-spatial” convexity algebras and the category of “m-sober” convexity spaces. We finally argue that the former notion of a point is more useful than the latter one from a category theoretic point of view and that the former notion of a point actually represents a polytope (or generic point) and the latter notion of a point properly represents a point. We also remark on the close relationships between pointfree convex geometry and domain theory.     

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)     

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.     

Separated totally convex spaces

In this paper we define for every totally convex space a suitable topology, the radial topology. We prove that a morphism in the category TC^sep of separated totally convex spaces is an epimorphism if and only if its image is dense in the radial topology, and that TC^sep is the full subcategory of TC generated by its Hausdorff objects. These results remain valid for finitely totally convex spaces when the radial topology is replaced by the distance-radial topology.Dedicated to Karl Stein     

Weakly compact multipliers on group algebras

We study left multipliers on the second dual spaces L¹(G)″ and M(G)″. We answer a question of Ghahramani and Lau, showing that for non-compact G a non-zero left multiplier on these spaces cannot be weakly compact.     

PROJECTIONS ON OPERATOR DUALS

Abstract

We discuss the existence of a projection with kernel K_b(E,F) ¹ (the annihilator of the quasi-compact operators) on the dual space of the space L _b,(E, F) of continous linear operators. Our results are proved in the context of Hausdorff locally convex spaces, but also provide extensions of recent results in the context of Banach spaces.     

Every Banach Space is Reflexive

The title above is wrong, because the strong dual of a Banach space is too strong to assert that the natural correspondence between a space and its bidual is an isomorphism. However, for many applications it suffices to replace the norm on the first dual by the weak*-structure in order to solve the non-reflexiveness problem [1]. But in this way, only the original vector space is recovered by taking the second dual. In this work we introduce a suitable numerical structure on vector spaces such that Banach balls, or more precisely totally convex modules, arise naturally in duality, namely as a category of Eilenberg–Moore algebras. This numerical structure naturally overlies the weak*-topology on the algebraic dual, so the entire Banach space can be reconstructed as a second dual. Moreover, the isomorphism between the original space and its bidual is the unit of an adjunction between the two-dualisation functors. Notice that the weak*-topology is normable only if it lives on a finite dimensional space; in that case the original space is trivial as well, hence reflexive. So the overlying numerical structure should be something more general than a norm or a seminorm and thus approach theory [2, 3] enters the picture.     

Traces of functions in Hardy and Besov spaces on Lipschitz domains with applications to compensated compactness and the theory of Hardy and Bergman type spaces

In this paper we study conditions guaranteeing that functions defined on a Lipschitz domain Ω have boundary traces in Hardy and Besov spaces on ∂Ω. In turn these results are used to develop a new approach to the theory of compensated compactness and the theory of non-locally convex Hardy and Bergman type spaces.     

Uniform convexity of Musielak-Orlicz-Sobolev spaces and applications

This paper deals with uniform convexity of Musielak-Orlicz-Sobolev spaces and its applications to variational problems. Some sufficient conditions and examples for uniform convexity of Musielak-Orlicz-Sobolev spaces are given. Some special properties relative to the uniformly convex modular for uniformly convex Musielak-Orlicz-Sobolev spaces are presented. As an application of these abstract results, the local minimizers and the mountain pass type critical point of an integral functional with more complicated growth than the p(x)-growth are studied.     

Average locally uniform rotundity and a class of nonlinear maps

We consider some topological characterizations of dual Banach spaces that admit an equivalent dual average locally uniformly rotund norm and provide a criterion for such renorming which involves the class of σ-slicely continuous maps.     

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.     

A cogenerator for preseparated superconvex spaces

We construct a cogenerator for the category of preseparated superconvex spaces, and we describe separated convex spaces, i.e. convex spaces for which the morphisms into the unit interval separates points.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

A dichotomy for the convex spaces of probability measures

We show that every nonempty compact and convex space M of probability Radon measures either contains a measure which has ‘small’ local character in M or else M contains a measure of ‘large’ Maharam type. Such a dichotomy is related to several results on Radon measures on compact spaces and to some properties of Banach spaces of continuous functions.