On compact multipliers of topological algebras

It is shown that if the maximal ideal space (A) of a semisimple commutative complete metrizable locally convex algebra contains no isolated points, then every compact multiplies is trivial. In particular, compact multipliers on semisimple commutative Fréchet algebras whose maximal ideal space has no isolated points are identically zero.     

Boundedly Compatible Webs and Strict Mackey Convergence

We look for characterizations of those locally convex spaces that satisfy the strict Mackey convergence condition within the context of spaces with webs. We will say that a locally convex space has a boundedly compatible web if it has a web of absolutely convex sets whose members behave like zero neighborhoods in a metrizable locally convex space. It will be shown that these locally convex spaces satisfy the strict Mackey convergence condition. One consequence of this result will be a characterization of boundedly retractive inductive limits. We will also prove that if E is locally complete and webbed, then the strict Mackey convergence condition is equivalent to E having a boundedly compatible web.     

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 theorem of Riesz type with Pettis integrals in topological vector spaces

Let X be a locally convex Hausdorff space and let C₀(S,X) be the space of all continuous functions f:S→X, with compact support on the locally compact space S. In this paper we prove a Riesz representation theorem for a class of bounded operators T:C₀(S,X)→X, where the representing integrals are X-valued Pettis integrals with respect to bounded signed measures on S. Under the additional assumption that X is a locally convex space, having the convex compactness property, or either, X is a locally convex space whose dual X^′ is a barrelled space for an appropriate topology, we obtain a complete identification between all X-valued Pettis integrals on S and the bounded operators T:C₀(S,X)→X they represent. Finally we give two illustrations of the representation theorem proved, in the particular case when X is the topological dual of a locally convex space.     

Strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms

Abstract

This article is devoted to C² a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature equation under the assumption of a strictly locally convex subsolution.     

Convex-compact sets and Banach discs

Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual E′ of a locally convex space E is the σ(E′,E)-closure of the union of countably many σ(E′,E)-relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.     

Convergence on closed convex sets in a locally convex space

The purpose of this paper is to compare several kinds of convergences on the space C(X) of nonempty closed convex subsets of a locally convex space X. First we verify that the AW-convergence on C(X) is weaker than the metric Attouch-Wets convergence on C(X) of a metrizable locally convex space X. Moreover, we show that X is normable if and only if the two convergences on C(X × R) are equivalent. Secondly we define two convergences on C(X) analogous to the corresponding ones in a normed linear space, and investigate some basic properties of these convergences and compare them.     

Some generalizations of locally and weakly locally uniformly convex space

In this paper, we prove that strongly convex space and almost locally uniformly rotund space, very convex space and weakly almost locally uniformly rotund space are respectively equivalent. We also investigate a few properties of k-strongly convex space and k-very convex space, and discuss the applications of strongly convex space and very convex space in approximation theory.     

Bounded approximation properties in non‐archimedean Banach spaces

Some non‐archimedean bounded approximation properties are introduced and studied in this paper. As an application, an affirmative answer is given, for non‐spherically complete base fields, to the following problem, posed in 13 , p. 95: Does there exist an absolutely convex edged set B in a non‐archimedean locally convex space such that its closure $\overline{B}Some non‐archimedean bounded approximation properties are introduced and studied in this paper. As an application, an affirmative answer is given, for non‐spherically complete base fields, to the following problem, posed in 13 , p. 95: Does there exist an absolutely convex edged set B in a non‐archimedean locally convex space such that its closure $\overline{B}$ is not edged?     

Varieties and Vector Measures

A locally convex space L has the property ? if equicontinuous subsets of L* are weak-star sequentially compact. (L*, σ(L*, L)) is a MAZUR space if given F ∈ L** with F weak-star sequentially continuous then F∈ L. If L is complete with the property ∈, then (L*, σ (L*, L)) is a MAZUR space. The class of locally convex spaces with the property ? forms a variety ??_? and this variety is generated by the BANACH spaces it contains. Weakly compactly generated locally convex spaces and SCHWARTZ spaces belong to ??_?. MAZUR spaces are used to give a characterization of GROTHENDIECK BANACH spaces. The last section contains a characterization of the variety generated by the reflexive BANACH spaces.     

Strict Topologies as Topological Algebras

Let X be a completely regular Hausdorff space, C_b(X) the space of all scalar-valued bounded continuous functions on X with strict topologies. We prove that these are locally convex topological algebras with jointly continuous multiplication. Also we find the necessary and sufficient conditions for these algebras to be locally m-convex.     

The injective hull op an operator ideal on locally convex spaces

In his book (3), Pietsch presents the problem of a direct construction of the injective hull of an operator ideal on loca. lly convex spaces. We construct the infective hull of an arbitrary operator ideal on locally convex spaces using the functor, where and B cover the family of all equicontinuous subsets of E'(L is the category of all locally convex spaces). If the operator ideal is bounded, we ob tain its infective hull using seminorm ideals.     

Contractibility and Connectedness of Efficient Point Sets

Using the technique of space theory and set-valued analysis, we establish contractibility results for efficient point sets in a locally convex space and a path connectedness result for a positive proper efficient point set in a reflexive space. We also prove a connectedness result for a positive proper efficient point set in a locally convex space; as an application, we give a connectedness result for an efficient solution set in a locally convex space.     

Convergence in probability of random power series and a related problem in linear topological spaces

A linear topological space is said to have the circle property if every power series with coefficients in it has a circle of convergence. Every complete locally convex or locally bounded space has the circle property, but not a certain class ofF-spaces including the space of all random variables on a non-atomic probability space, endowed with the topology of convergence in probability. Research sponsored by the National Science Foundation under Grant No. GP 6035.     

Banach Disks and Marrelledness Properties of Metrizalbe <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript> (Ω, <Emphasis Type="Italic">X</Emphasis>)

Let be a non-empty set and X a metrizable locally convex space. We show that the metrizable locally convex space c₀ (, X) is p-barrelled (totally barrelled) if and only if X is p-barrelled (totally barrelled). Some applications for closed graph theorems are included.     

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     

Positivity aspects of the Fantappiè transform

A Hilbert space approach to the classical Fantappiè transform, based on the concept of Gel'fand triples of locally convex spaces, leads to a novel proof of Martineau-Aizenberg duality theorem. A study of Fantappiè transforms of positive measures on the unit ball inC ⁿ relates ideas of realization theory of multivariate linear systems, locally convex duality and pluripotential theory. This is applied to obtain von Neumann type estimates on the joint numerical range of tuples of Hilbert space operators. Partially supported by National Science Foundation Grants DMS 0070639 and DMS 0322255. Partially supported by National Science Foundation Grant DMS 0100367.     

Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

ON THE PRODUCT OF GATEAUX DIFFERENTIABILITY LOCALLY CONVEX SPACES

A locally convex space is said to be a Gâteaux differentiability space (GDS) provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gâteaux differentiable in D. This paper shows that the product of a GDS and a family of separable Fréchet spaces is a GDS, and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS.     

Uniquely generated Grothendieck space ideals

Starting from a result of Pietsch on the Grothendieck ideal of strictly nuclear locally convex spaces, we classify all those Grothendieck ideals of nuclear locally convex spaces that are generated by a unique operator ideal on the class of Banach spaces.