Real Topological Vector Space Research Articles

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.

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.

A generalization of interval-valued optimization problems and optimality conditions by using scalarization and subdifferentials

In this work, interval-valued optimization problems are considered. The ordering cone is used to generalize the interval-valued optimization problems on real topological vector spaces. Some definitions and their properties are obtained for intervals, defined via an ordering cone. Gerstewitz's function is used to derive scalarization for the interval-valued optimization problems. Also, two subdifferentials for interval-valued functions are introduced by using subgradients. Some necessary optimality conditions are obtained via subdifferentials and scalarization. An example is given to demonstrate the results.

Nonconvex Equilibrium Problems Via a KKM Theorem

In this paper, we establish existence results for vector equilibrium problems without convexity assumptions on the whole domain, but just on the closure of a quasi-self-segment-dense subset. In the absence of demanding technical assumptions, we present new conditions to obtain results on the existence of solutions of nonconvex equilibrium problems in real topological vector spaces via a KKM theorem. Furthermore, no coercivity conditions are used to deal with the noncompact case in reflexive Banach spaces. Examples are provided to compare the obtained results with the literature in convex and nonconvex cases.

Stability of efficient solutions to set optimization problems

This article deals with considering stability properties of Pareto minimal solutions to set optimization problems with the set less order relation in real topological Hausdorff vector spaces. We focus on studying the Painleve–Kuratowski convergence of Pareto minimal elements in the image space. Employing convexity properties, we study the external stability of Pareto minimal solutions via weak ones. Then, we use converse properties to investigate external stability conditions to such problems where Pareto minimal solution sets and weak/ideal ones are distinct. For the internal stability, we propose a concept of compact convergence in the sense of Painleve–Kuratowski and use it together with a domination property to analyze stability conditions for the reference problems.

Decision Making and Games with Vector Outcomes

Abstract In this paper, we study decision making and games with vector outcomes. We provide a general framework where outcomes lie in a real topological vector space and the decision maker’s preferences over outcomes are described by a preference cone, which is defined as a convex cone satisfying a continuity axiom. Further, we define a notion of utility representation and introduce a duality between outcomes and utilities. We provide conditions under which a preference cone is represented by a utility and is the dual of a set of utilities. We formulate a decision-making problem with vector outcomes and study optimal choices. We also consider games with vector outcomes and characterize the set of equilibria. Lastly, we discuss the problem of equilibrium selection based on our characterization.

On the Mazur–Ulam Theorem

We give an extension of the Mazur–Ulam theorem for isometries from real metrizable topological vector spaces into real normed spaces.

Scalarizations for a unified vector optimization problem based on order representing and order preserving properties

The aim of this paper is to study characterizations of minimal and approximate minimal solutions for a unified vector optimization problem in a Hausdorff real topological vector space. These characterizations have been obtained via scalarizations which are based on general order representing and order preserving properties. A nonlinear scalarization based on Gerstewitz function is shown to be a particular case of the proposed scalarizations. Furthermore, in the setting of normed space, characterizations are given for minimal solutions by using scalarization function based on the oriented distance function. Finally, under appropriate assumptions it is shown that this function satisfies order representing and order preserving properties.

Q-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form $$C_{R,T}(S)=RST$$ to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-frequent hypercyclicity criterion, then the map $$C_{R}(S)=RSR^*$$ is shown to be q-frequently hypercyclic on the space $$\mathcal {K}(H)$$ of all compact operators and the real topological vector space $$\mathcal {S}(H)$$ of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for $$C_{R,T}$$ to be q-frequently hypercyclic on the Schatten von Neumann classes $$S_p(H)$$ . We also characterize frequent hypercyclicity of $$C_{M^*_\varphi ,M_\psi }$$ on the trace-class of the Hardy space, where the symbol $$M_\varphi $$ denotes the multiplication operator associated to $$\varphi $$ .

Global optimization of the difference of affine increasing and positively homogeneous functions

The theory of increasing and positively homogeneous (IPH) functions defined on a real topological vector space X has well been developed. In this paper, we first give various characterizations for maximal elements of the support set of this class of functions. As an application, we present various characterizations for maximal elements of the support set of affine IPH functions. Finally, we investigate necessary and sufficient conditions for the global minimum of the difference of two strictly affine IPH functions.

Additive iterative roots of identity and Hamel bases

We describe a class of discontinuous additive functions \({a:X\to X}\) on a real topological vector space X such that \({a^n={\rm id}_X}\) and \({a({\mathcal{H}}){\setminus} {\mathcal{H}} \neq\emptyset}\) for every infinite set \({{\mathcal{H}} \subset X}\) of vectors linearly independent over \({\mathbb{Q}}\). We prove the density of the family of all such functions in the linear topological space \({{\mathcal{A}}_X}\) of all additive functions \({a:X\to X}\) with the topology induced on \({{\mathcal{A}}_X}\) by the Tychonoff topology of the space X X . Moreover, we consider additive functions \({a\in{\mathcal{A}}_X}\) satisfying \({a^n={\rm id}_X}\) and \({a({\mathcal{H}})= {\mathcal{H}}}\) for some Hamel basis \({{\mathcal{H}}}\) of X. We show that the class of all such functions is also dense in \({{\mathcal{A}}_X}\). The method is based on decomposition theorems for linear endomorphisms.

On i-operators on real normed spaces

We gradually study i-operators on real vector spaces, on real topological vector spaces, and on real normed spaces. Among several things we prove the existence of real topological vector spaces (different from the James’ space) that are free of continuous i-operators. We also prove that every real normed space can be equivalently renormed to be free of norm i-operators. Examples of spaces of continuous functions not admitting norm i-operators and whose unit sphere is free of convex subsets with non-empty interior relative to it are also found. Finally, we also provide some results on a problem posed by Wenzel in [10].

Convex components and multi-slices in real topological vector spaces

It is shown that, in a non-necessarily Hausdorff real topological vector space, if a subset is a countable disjoint union of convex sets closed in the subset, then those convex sets must be its convex components. On the other hand, by means of convex components we extend the notion of extreme point to non-convex sets, which entails a new equivalent reformulation of the Krein--Milman property (involving drops among other objects). Finally, we study the nature of convex functions and provide some results on their support in order to introduce the concept of multi-slice, that is, slices determined by convex functions (instead of by linear functions). Among other things, we prove that the boundary of a closed convex set with non-empty interior can be obtained as the set of support points of a certain lower semi-continuous convex function on that convex set.

Existence of Solutions to Weakly Generalized Vector F-implicit Variational Inequalities

In this paper we studied a new class of weakly generalized vector F-implicit variational inequalities in real topological vector spaces. We use the Ferro minimax theorem to discuss the existence of strong solutions for generalized vector F-implicit variational inequalities.

On some vector ƒ-complementarity problems

Recently, Huang, Li and O'Regan introduced a class of (scalar) generalized -complementarity problems and three classes of variational inequalities in real Banach spaces for a fixed cone. Our aim is to introduce a vector case of their problems in the real topological vector spaces setting for a moving cone and to generalize their results even in the scalar case.

On the Krein–Milman Property and the Bade Property

Throughout this paper a study on the Krein–Milmam Property and the Bade Property is entailed reaching the following conclusions: If a real topological vector space satisfies the Krein–Milmam Property, then it is Hausdorff; if a real topological vector space satisfies the Krein–Milmam Property and is locally convex and metrizable, then all of its closed infinite dimensional vector subspaces have uncountable dimension; if a real pseudo-normed space has the Bade Property, then it is Hausdorff as well but could allow closed infinite dimensional vector subspaces with countable dimension. On other hand, we show the existence of infinite dimensional closed subspaces of ℓ∞ with the Bade Property that are not the space of convergence associated to any series in a real topological vector space. Finally, we characterize unconditionally convergent series in real Banach spaces by means of a new concept called uniform convergence of series.

On F-Implicit Generalized Vector Variational Inequalities

In this paper, we study the F-implicit generalized (weak) case for vector variational inequalities in real topological vector spaces. Both weak and strong solutions are considered. These two sets of solutions coincide whenever the mapping T is single-valued, but not set-valued. We use the Ferro minimax theorem to discuss the existence of strong solutions for F-implicit generalized vector variational inequalities.

Linearization of certain uniform homeomorphisms

AbstractThis article concerns the uniform classification of infinite dimensional real topological vector spaces. We examine a recently isolated linearization procedure for uniform homeomorphisms of the form φ: X →Y, where X is a Banach space with non-trivial type and Y is any topological vector space. For such a uniform homeomorphism φ, we show that Y must be normable and have the same supremal type as X. That Y is normable generalizes theorems of Bessaga and Enflo. This aspect of the theory determines new examples of uniform non-equivalence. That supremal type is a uniform invariant for Banach spaces is essentially due to Ribe. Our linearization approach gives an interesting new proof of Ribe's result.

Hölder continuous homomorphisms between infinite-dimensional Lie groups are smooth

Let f : G → H be a homomorphism between smooth Lie groups modelled on Mackey complete, locally convex real topological vector spaces. We show that if f is Hölder continuous at 1, then f is smooth.

Vector Variational Inequalities and the (S)+ Condition

Let $${\cal Z}$$ and X be Hausdorff real topological vector spaces and let $${\cal L}_b(X,{\cal Z})$$ be the space of continuous linear mappings from X into $${\cal Z}$$ equipped with the topology of bounded convergence. In this paper, we define the (S)+ condition for operators from a nonempty subset of X into $${\cal L}_b(X,{\cal Z})$$ and derive some existence results for vector variational inequalities with operators of the class (S)+. Some applications to vector complementarity problems are given.