Locally Convex Spaces Research Articles

Is the Free Locally Convex Space L(X) Nuclear?

Given a class $${{\mathcal {P}}}$$ of Banach spaces, a locally convex space (LCS) E is called multi- $${{\mathcal {P}}}$$ if E can be isomorphically embedded into a product of spaces that belong to $${{\mathcal {P}}}$$ . We investigate the question whether the free locally convex space L(X) is strongly nuclear, nuclear, Schwartz, multi-Hilbert or multi-reflexive. If X is a Tychonoff space containing an infinite compact subset then, as it follows from the results of Außenhofer (Topol Appl 134:90–102, 2007), L(X) is not nuclear. We prove that for such X the free LCS L(X) has the stronger property of not being multi-Hilbert. We deduce that if X is a k-space, then the following properties are equivalent: (1) L(X) is strongly nuclear; (2) L(X) is nuclear; (3) L(X) is multi-Hilbert; (4) X is countable and discrete. On the other hand, we show that L(X) is strongly nuclear for every projectively countable P-space (in particular, for every Lindelöf P-space) X. We observe that every Schwartz LCS is multi-reflexive. It is known that if X is a $$k_\omega $$ -space, then L(X) is a Schwartz LCS (Außenhofer et al. in Stud Math 181(3):199–210, 2007), hence L(X) is multi-reflexive. We show that for every first-countable paracompact (in particular, for every metrizable) space X the converse is true, so L(X) is multi-reflexive if and only if X is a $$k_\omega $$ -space, equivalently if X is a locally compact and $$\sigma $$ -compact space. Similarly, we show that for any first-countable paracompact space X the free abelian topological group A(X) is a Schwartz group if and only if X is a locally compact space such that the set $$X^{(1)}$$ of all non-isolated points of X is $$\sigma $$ -compact.

Fixed point theorems for countably asymptotically Ф-nonexpansive maps in locally convex spaces and application

In this paper, we introduce the concept of a countably asymptotically ?-nonexpansive operator. In addition, we establish new fixed point results for some countably asymptotically ?-nonexpansive and sequentially continuous maps, fixed-point results of Krasnosel?skii type in locally convex spaces. Moreover, we present Leray-Schauder-type fixed point theorems for countably asymptotically ?-nonexpansive maps in locally convex spaces. Apart from that we show the applicability of our results to the theory of Volterra integral equations in locally convex spaces. The main condition in our results is formulated in terms of the axiomatic measure of noncompactness. Our results improve and extend in a broad sense recent ones obtained in literature.

K-spaces and duals of non-archimedean metrizable locally convex spaces

Abstract The main purpose of this paper is to investigate the non-archimedean counterpart of the classical result stating that the dual of a real or complex metrizable locally convex space, equipped with the locally convex topology of uniform convergence on compact sets, belongs to the topological category formed by the k-spaces. We prove that this counterpart holds when the non-archimedean valued base field 𝕂 {\mathbb{K}} is locally compact, but fails for any non-locally compact 𝕂 {\mathbb{K}} . Here we deal with a topological subcategory, the one formed by the k 0 {k_{0}} -spaces, the adequate non-archimedean substitutes for k-spaces. As a product, we complete some of the achievements on the non-archimedean Banach–Dieudonné Theorem presented in [C. Perez-Garcia and W. H. Schikhof, The p-adic Banach–Dieudonné theorem and semi-compact inductive limits, p-adic Functional Analysis (Poznań 1998), Lecture Notes Pure Appl. Math. 207, Dekker, New York 1999, 295–307]. Also, we use our results to construct in a simple way natural examples of k-spaces (which are also k 0 {k_{0}} -spaces) whose products are not k 0 {k_{0}} -spaces. This in turn improves the, rather involved, example given in [C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over non-Archimedean Valued Fields, Cambridge Stud. Adv. Math. 119, Cambridge University Press, Cambridge, 2010] of two k 0 {k_{0}} -spaces whose product is not a k 0 {k_{0}} -space. Our theory covers an important class of non-archimedean Fréchet spaces, the Köthe sequence spaces, which have a relevant influence on applications such as the definition of a non-archimedean Laplace and Fourier transform.

Differential forms on locally convex spaces and the stokes formula

In this paper we prove a variant of the Stokes formula for differential forms of a finite codimension in a locally convex space (LCS). The main tool used by us for proving the mentioned formula is the surface layer theorem for surfaces of codimension 1 in a locally convex space which was proved earlier by the first author. Moreover, on some subspace of differential forms of the Sobolev type with respect to a differentiable measure we establish a formula expressing the operator adjoint to the exterior differential via standard operations of the calculus of differential forms and the logarithmic derivative. This connection was established earlier under stronger constraints imposed either on the LCS or on the measure, or on differential forms (the smoothness condition).

Hilbert supports of measures on locally convex spaces

In the paper, we discuss relationships between the existence of Hilbert supports of countably additive measures on a locally convex space (LCS) and the continuity of their Fourier transforms in the Gross–Sazonov topology (which is defined below) on the dual space. In particular, it follows from the theorem proved in the paper that the Fourier transform of the Wiener measure on C[0, 1] is not continuous in the Gross–Sazonov topology on the dual space \({\left( {C\left[ {0,1} \right]} \right)^\prime }\) endowed with the Mackey topology (the strongest locally convex topology among those consistent with the duality between \({\left( {C\left[ {0,1} \right]} \right)^\prime }\) and C[0, 1]).

Common Fixed Point Theorem For Uniformly R-subweakly Commuting Mappings In Hausdorff Locally Convex Space

We show common fixed point theorem for generalized non-expansive and uniformly R-subweakly commuting mappings satisfying a more uniformly asymptotically regular condition in Hausdorff locally convex space.

GRADUAL NORMED LINEAR SPACE

In this paper, the gradual real numbers are considered and the notion of the gradual normed linear space is given. Also some topological properties of such spaces are studied, and it is shown that the gradual normed linear space is a locally convex space, in classical sense. So the results in locally convex spaces can be translated in gradual normed linear spaces. Finally, we give an example of a gradual normed linear space which is not normable in classical analysis.

ON THE PRINCIPLE OF UNIFORM BOUNDEDNESS FOR LSC CONVEX PROCESSES IN A STRICTLY N LOCALLY CONVEX SPACES

Article ON THE PRINCIPLE OF UNIFORM BOUNDEDNESS FOR LSC CONVEX PROCESSES IN A STRICTLY N LOCALLY CONVEX SPACES was published on January 1, 2008 in the journal Demonstratio Mathematica (volume 41, issue 1).

Superefficiency in Local Convex Spaces

In the framework of normed spaces, Borwein and Zhuang introduced superefficiency and gave its concise dual form when the underlying decision problem is convex. In this paper, we consider four different generalizations of the Borwein and Zhuang superefficiency in locally convex spaces and give their concise dual forms for convex vector optimization. When the ordering cone has a base, we clarify the relationship between Henig efficiency and the various kinds of superefficiency. Finally, we show that whether the four kinds of superefficiency are equivalent to each other depends on the normability of the underlying locally convex spaces.

Local convexity, adapted experiments and the wave equation: Lecam's randomization criterion and related results

We provide generalizations of LeCam's randomization criterion and present related results which we call black-box criteria. We investigate the general functional analytic structure behind randomization criteria and black-box criteria. We show by a general categorical argument that randomization criteria and corresponding black-box criteria are equivalent. By abstraction to locally convex spaces we eliminate the lattice structure usually involved. The applications of our general functional analytic results range from the comparison of adapted statistical experiments to the wave equation.

Full Nuclear Cones and a Relation Between Strong Optimization and Pareto Efficiency

In this paper we present some new and pertinent connections between the Strong Optimization and the Approximate Pareto type Efficiency, in particular, with the usual Vector Optimization, at first in the Ordered Vector Spaces by the natural Convex Cones and, afterwards, in the Ordered Hausdorff Locally Convex Spaces. The main result is obtained considering the notion of full nuclear cone. Our results, is related to an appropriate scalarization method

Multiplication Operators and Dynamical Systems on Weighted Locally Convex Spaces of Holomorphic Functions

Abstract Let 𝐺 be an open subset of ℂ and let 𝑉 be an arbitrary system of weights on 𝐺. Let 𝐻𝑉𝑏(𝐺) and 𝐻𝑉0(𝐺) be the weighted locally convex spaces of holomorphic functions with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present article, we characterize the analytic functions inducing multiplication operators and invertible multiplication operators on the spaces 𝐻𝑉𝑏(𝐺) and 𝐻𝑉0(𝐺) for different systems of weights 𝑉 on 𝐺. A (linear) dynamical system induced by multiplication operators on these spaces is obtained as an application of the theory of multiplication operators.

SEQUENTIALLY CONTINUOUS NON-LINEAR FUNDAMENTAL SYSTEMS OF SOLUTIONS OF AFFINE EQUATIONS IN LOCALLY CONVEX SPACES

Article SEQUENTIALLY CONTINUOUS NON-LINEAR FUNDAMENTAL SYSTEMS OF SOLUTIONS OF AFFINE EQUATIONS IN LOCALLY CONVEX SPACES was published on July 1, 2003 in the journal Demonstratio Mathematica (volume 36, issue 3).

A characterization of Pontryagin–van Kampen duality for locally convex spaces

Topological vector spaces (TVSs) are topological Abelian groups when considered under the operation of addition. It is therefore natural to ask when they satisfy Pontryagin–van Kampen (P–vK) duality. In 1984 S. Kye published a characterization of P–vK duality for real locally convex spaces (LCSs). His proof however is incorrect. In this paper we offer an alternative characterization of P–vK duality for real LCSs. We also compare our results with some other contributions and state a number of questions.

The Free Abelian Topological Group and the Free Locally Convex Space on the Unit Interval

We give a complete description of the topological spaces X such that the free abelian topological group A(X) embeds into the free abelian topological group A(I) on the closed unit interval. In particular, the free abelian topological group A(X) on any finite-dimensional compact metrizable space X embeds into A(I). To obtain our description, we study similar embeddings of the free locally convex spaces and continuous surjections between the spaces of continuous functions with the pointwise topology. Proofs are based on the classical Kolmogorov's Superposition Theorem.

Topological decompositions of the duals of locally convex operator spaces

If Z and E are Hausdorff locally convex spaces (LCS) then by Lb(Z, E) we mean the space of continuous linear maps from Z to E endowed with the topology of uniform convergence on the bounded subsets of Z. The dual Lb(Z, E)′ will always carry the topology of uniform convergence on the bounded subsets of Lb(Z, E). If K(Z, E) is a linear subspace of L(Z, E) then Kb(Z, E) will be used to denote K(Z, E) with the relative topology and Kb(Z, E)″ will mean the dual of Kb(Z, E)′ with the natural topology of uniform convergence on the equicontinuous subsets of Kb(Z, E)′. If Z and E are Banach spaces these provide, in each instance, the usual norm topologies.

A Free Subgroup of the Free Abelian Topological Group on the unit Interval

Bulletin of the London Mathematical SocietyVolume 14, Issue 5 p. 399-402 Notes and papers A Free Subgroup of the Free Abelian Topological Group on the unit Interval Eli Katz, Eli Katz Cleveland State University, Cleveland Ohio 44115, U.S.A.Search for more papers by this authorSidney A. Morris, Sidney A. Morris La Trobe University, Bundoora Victoria 3083, AustraliaSearch for more papers by this authorPeter Nickolas, Peter Nickolas University of Queensland, St. Lucia, Queensland 4067, AustraliaSearch for more papers by this author Eli Katz, Eli Katz Cleveland State University, Cleveland Ohio 44115, U.S.A.Search for more papers by this authorSidney A. Morris, Sidney A. Morris La Trobe University, Bundoora Victoria 3083, AustraliaSearch for more papers by this authorPeter Nickolas, Peter Nickolas University of Queensland, St. Lucia, Queensland 4067, AustraliaSearch for more papers by this author First published: September 1982 https://doi.org/10.1112/blms/14.5.399Citations: 10AboutRelatedInformationPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessClose modalShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume14, Issue5September 1982Pages 399-402 RelatedInformation RecommendedThe Free Abelian Topological Group and the Free Locally Convex Space on the Unit IntervalArkady Leiderman, Sidney A. Morris, Vladimir Pestov, Journal of the London Mathematical SocietyTopological properties of spaces admitting free group actionsRoss Geoghegan, Craig R. Guilbault, Journal of TopologyEMBEDDING OF THE FREE ABELIAN TOPOLOGICAL GROUP INTO Mikołaj Krupski, Arkady Leiderman, Sidney Morris, MathematikaKulikov's Problem on Universal Torsion-Free Abelian GroupsSaharon Shelah, Lutz Strüngmann, Journal of the London Mathematical SocietyGroup Laws and Free Subgroups in Topological GroupsMiklós Abért, Bulletin of the London Mathematical Society

The range of a vector measure has the Banach-Saks property

The above result of Diestel and Seifert is proved using the Banach-Saks theorem for L p ( λ ) , 1 > p > ∞ {L^p}(\lambda ),1 > p > \infty .

Decomposition of Precompact Operators in Ordered Locally Convex Spaces

Decomposition of Precompact Operators in Ordered Locally Convex Spaces

Full-Completeness in Weighted Spaces

That the class of fully complete (B-complete) locally convex spaces has an essential role has been well-established (see, e.g., Collins [2], Husain [12], and Pták [16]). However, it is still the case that very little is known concerning full-completeness in function spaces except when implied by formally stronger properties (e.g., Fréchet), and our aim in this article is to shed some light in this direction. Our approach is to use the unifying notion of the weighted spaces CV0(X) discussed in [19; 20], since such spaces provide a general setting for the study of virtually all continuous function spaces encountered in analysis.The results to be presented here divide naturally into three sections with § 2 being consigned the preliminary definitions and theorems. In § 3 we obtain a necessary condition for full-completeness in a class of function spaces, and consider certain consequences of this result.