Convex Topological Vector Space Research Articles

Properties of generalized polyhedral convex multifunctions

This paper presents a study of generalized polyhedral convexity under basic operations on multifunctions. We address the preservation of generalized polyhedral convexity under sums and compositions of multifunctions, the domains and ranges of generalized polyhedral convex multifunctions, and the direct and inverse images of sets under such mappings. Then we explore the class of optimal value functions defined by a generalized polyhedral convex objective function and a generalized polyhedral convex constrained mapping. The new results provide a framework for representing the relative interior of the graph of a generalized polyhedral convex multifunction in terms of the relative interiors of its domain and mapping values in locally convex topological vector spaces. Among the new results in this paper is a significant extension of a result by Bonnans and Shapiro on the domain of generalized polyhedral convex multifunctions from Banach spaces to locally convex topological vector spaces.

Mosco Convergence of Gradient Forms with Non-Convex Interaction Potential

This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, EN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {E}}^N$$\\end{document} on L2(E,μN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(E,\\mu _N)$$\\end{document} for N∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\in {\\mathbb {N}}$$\\end{document}, in the framework of converging Hilbert spaces by K. Kuwae and T. Shioya. The basic assumption is weak measure convergence of the family (μN)N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${(\\mu _N)}_{N}$$\\end{document} on the state space E—either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on (μN)N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${(\\mu _N)}_{N}$$\\end{document} try to impose as little restrictions as possible. The problem has fully been solved if the family (μN)N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${(\\mu _N)}_{N}$$\\end{document} contain only log-concave measures, due to Ambrosio et al. (Probab Theory Relat. Fields 145:517–564, 2009). However, for a large class of convergence problems the assumption of log-concavity fails. The article suggests a way to overcome this hindrance, as it presents a new approach. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result by Bounebache and Zambotti (J Theor Probab 27:168–201, 2014).

Carathéodory-Hahn-Kluvánek extension theorem in locally convex spaces with application

In this paper, we present a Carathéodory-Hahn-Kluvánek-type theorem for a multimeasure H:A→cwk(X) on A, where A is an algebra of subsets of a set Ω and cwk(X) is the family of all convex weakly compact nonempty subsets of a non-separable locally convex topological vector space X. Applying extension results for additive multimeasures, it will be proved a Radon-Nikodým theorem for an additive multimeasure in terms of “nonempty weakly density”.

Characterizations of minimal elements of upper support with applications in minimizing DC functions

Abstract In this study, we discuss on the problem of minimizing the differences of two non-positive valued increasing, co-radiant and quasi-concave (ICRQC) functions defined on X X (where X X is a real locally convex topological vector space). For this purpose, we first gave different characterizations of the upper support set’s minimal elements of non-positive co-radiant functions. Then, we presented sufficient and necessary conditions for the global minimizers of the differences of two non-positive ICRQC functions.

Generalizations of Rolle’s Theorem

The classical Rolle’s theorem establishes the existence of (at least) one zero of the derivative of a continuous one-variable function on a compact interval in the real line, which attains the same value at the extremes, and it is differentiable in the interior of the interval. In this paper, we generalize the statement in four ways. First, we provide a version for functions whose domain is in a locally convex topological Hausdorff vector space, which can possibly be infinite-dimensional. Then, we deal with the functions defined in a real interval: we consider the case of unbounded intervals, the case of functions endowed with a weak derivative, and, finally, we consider the case of distributions over an open interval in the real line.

Extension theorems of additive interval multifunctions with applications

In this paper, we present extension theorems of additive interval multifunctions to multimeasures in locally convex topological vector spaces. Applying these extension results, we present Radon-Nikodym theorems for additive interval multifunctions.

Generalized relative interiors and generalized convexity in infinite-dimensional spaces

This paper focuses on investigating generalized relative interior notions for sets in locally convex topological vector spaces with particular attentions to graphs of set-valued mappings and epigraphs of extended-real-valued functions. We introduce, study, and utilize a novel notion of quasi-near convexity of sets that is an infinite-dimensional extension of the widely acknowledged notion of near convexity. Quasi-near convexity is associated with the quasi-relative interior of sets, which is investigated in the paper together with other generalized relative interior notions for sets, not necessarily convex. In this way, we obtain new results on generalized relative interiors for graphs of set-valued mappings in convexity and generalized convexity settings.

A Denjoy-type integral and a strongly henstock integral in locally convex topological vector space

A Denjoy-type integral and a strongly henstock integral in locally convex topological vector space

Schauder-Tychonoff Fixed Point Theorem on Sequentially Complete Hausdorff Strongly Convex Topological Vector Spaces

In this paper, we study the Schauder-Tychonoff fixed point (STFP) on a subset A of a sequentially complete Hausdorff strongly convex topological vector space (SCHSCTVS) E (over the field R) with calibration Γ have a unique STFP in Topological Vector Space (TVS).

TOPOLOGICAL MODULES GEOMETRY

The extremal structure of zero-neighbourhoods of a topological module is analyzed reaching unexpected conclusions when the module topology is not Hausdorff. These results motivate us to introduce the notion of metric modules, which are modules endowed with a translation-invariant metric, turning them into an (additive) topological group. We study the central and diametral points of additively symmetric subsets and find examples of convex sets which are not symmetric translates (translates of addivitely symmetric subsets). As a consequence of all of these, it seems natural to transport the well-known Bishop-Phelps property from the category of real topological vector spaces to general topological modules over topological rings. Then we stick to particular topological rings, the unital C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras, showing that the subset of positive elements lying below the unity is an effect algebra. We also prove that every continuous linear operator on a Hausdorff locally convex topological vector space that commutes with all continuous linear projections of one-dimensional range is a multiple of the identity. Finally, we discuss how to transport the previous result to C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras.

Stability of linear operators in locally convex cones

The theory of locally convex cones generalizes locally convex topological vector spaces and provides many different examples and applications than locally convex vector spaces. The stability problem in the sense of Ulam has not yet received any attention in the theory of locally convex cones. Therefore, in this paper, we introduce the stability problem in locally convex cones and provide the stability of linear operators in locally convex cones.

Multipliers of Variationally Mcshane Integrable Functions in Locally Convex Space

Abstract We study measurable real valued multipliers of variationally McShane (resp. McShane) integrable functions defined on a σ-finite outer regular quasi-Radon measure space and taking values in a complete locally convex topological vector space X. We also show: in case X is representable by semi-norm then essentially bounded real measurable functions are multipliers of functions which are Pettis integrable as well as integrable by semi-norm. The space of real valued measurable and essentially bounded functions turn out to be precisely the multipliers of variationally McShane (resp. McShane) integrable functions in case X is representable by semi-norm.

On Bishop–Phelps and Krein–Milman Properties

A real topological vector space is said to have the Krein–Milman property if every bounded, closed, convex subset has an extreme point. In the case of every bounded, closed, convex subset is the closed convex hull of its extreme points, then we say that the topological vector space satisfies the strong Krein–Milman property. The strong Krein–Milman property trivially implies the Krein–Milman property. We provide a sufficient condition for these two properties to be equivalent in the class of Hausdorff locally convex real topological vector spaces. This sufficient condition is the Bishop–Phelps property, which we introduce for real topological vector spaces by means of uniform convergence linear topologies. We study the inheritance of the Bishop–Phelps property. Nontrivial examples of topological vector spaces failing the Krein–Milman property are also given, providing us with necessary conditions to assure that the Krein–Milman property is satisfied. Finally, a sufficient condition to assure the Krein–Milman property is discussed.

On semireflexive subcategories

In the category of Hausdorff locally convex topological vector spaces it is known that the class of L-semireflexive subcategories and some of its subclasses are isomorphic to classes of reflexive or coreflective subcategories. The most general case occurs when L is a c-reflective subcategory (L contains the subcategory of spaces with weak topology and the reflector functor is exactly to the left).This article examines a situation in which the coreflector functor commutes with products.

Weakly almost periodic functions invariant means and fixed point properties in locally convex topological vector spaces

Weakly almost periodic functions invariant means and fixed point properties in locally convex topological vector spaces

Generalized Differentiation and Duality in Infinite Dimensions under Polyhedral Convexity

This paper addresses the study and applications of polyhedral duality in locally convex topological vector (LCTV) spaces. We first revisit the classical Rockafellar’s proper separation theorem for two convex sets one of which is polyhedral and then present its LCTV extension replacing the relative interior by its quasi-relative interior counterpart. Then we apply this result to derive enhanced calculus rules for normals to convex sets, coderivatives of convex set-valued mappings, and subgradients of extended-real-valued functions under certain polyhedrality requirements in LCTV spaces by developing a geometric approach. We also establish in this way new results on conjugate calculus and duality in convex optimization with relaxed qualification conditions in polyhedral settings. Our developments contain significant improvements to a number of existing results obtained by Ng and Song (Nonlinear Anal. 55, 845–858, 12).

Inverse of Berge’s maximum theorem in locally convex topological vector spaces and its applications

Abstract In this paper, the inverse of Berge’s maximum theorem is established in a locally convex topological vector space. Using this result, the generalized Gale–Nikaido–Debreu’s lemma and the generalized coincidence point theorem are derived from the equilibrium theorem of generalized games. By combining the inverse of Berge’s maximum theorem with the generalized coincidence point theorem, the equilibrium theorem of the generalized game is obtained immediately. Finally, we show that Fan–Glicksberg’s fixed point theorem and von Neumann’s lemma can be derived directly from the coincidence point theorem.

Constrained Discounted Stochastic Games

In this paper, we consider a large class of constrained non-cooperative stochastic Markov games with countable state spaces and discounted cost criteria. In one-player case, i.e., constrained discounted Markov decision models, it is possible to formulate a static optimisation problem whose solution determines a stationary optimal strategy (alias control or policy) in the dynamical infinite horizon model. This solution lies in the compact convex set of all occupation measures induced by strategies, defined on the set of state–action pairs. In case of n-person discounted games the occupation measures are induced by strategies of all players. Therefore, it is difficult to generalise the approach for constrained discounted Markov decision processes directly. It is not clear how to define the domain for the best-response correspondence whose fixed point induces a stationary equilibrium in the Markov game. This domain should be the Cartesian product of compact convex sets in a locally convex topological vector space. One of our main results shows how to overcome this difficulty and define a constrained non-cooperative static game whose Nash equilibrium induces a stationary Nash equilibrium in the Markov game. This is done for games with bounded cost functions and positive initial state distribution. An extension to a class of Markov games with unbounded costs and arbitrary initial state distribution relies on an approximation of the unbounded game by bounded ones with positive initial state distributions. In the unbounded case, we assume the uniform integrability of the discounted costs with respect to all probability measures induced by strategies of the players, defined on the space of plays (histories) of the game. Our assumptions are weaker than those applied in earlier works on discounted dynamic programming or stochastic games using the so-called weighted norm approach.

Fenchel–Rockafellar theorem in infinite dimensions via generalized relative interiors

In this paper we provide further studies of Fenchel duality theory in the general framework of locally convex topological vector (LCTV) spaces. We prove the validity of the Fenchel strong duality under some qualification conditions via generalized relative interiors imposed on the epigraphs and the domains of the functions involved. Our results directly generalize the classical Fenchel–Rockafellar theorem on strong duality from finite dimensions to LCTV spaces.

Open discrete shrinkings in function spaces

Given a sequence U={Un:n∈ω} of non-empty open subsets of a space X, a family {Vn:n∈ω} is a shrinking of U if Vn is a non-empty open set and Vn⊂Un for every n∈ω. A space X has discrete shrinking property if every sequence of non-empty open subsets of X has a discrete shrinking. We show that if L is a non-metrizable locally convex topological vector space and Lw is the set L with the weak topology of the space L, then Lw has the discrete shrinking property. In particular, a space X is uncountable if and only if Cp(X) has the discrete shrinking property. The same statement is not true for Cp(X,[0,1]) so we study when the discrete shrinking property holds in Cp(X,[0,1]). We prove, among other things, that Cp(X,[0,1]) features discrete shrinking property if X is an essentially uncountable space.