Topological Vector Spaces 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.

Generalized hyperbolicity, stability and expansivity for operators on locally convex spaces

We introduce and study the notions of (generalized) hyperbolicity, topological stability and (uniform) topological expansivity for operators on locally convex spaces. We prove that every generalized hyperbolic operator on a locally convex space has the finite shadowing property. Contrary to what happens in the Banach space setting, hyperbolic operators on Fréchet spaces may fail to have the shadowing property, but we find additional conditions that ensure the validity of the shadowing property. Assuming that the space is sequentially complete, we prove that generalized hyperbolicity implies the strict periodic shadowing property, but we also show that the hypothesis of sequential completeness is essential. We show that operators with the periodic shadowing property on topological vector spaces have other interesting dynamical behaviors, including the fact that the restriction of such an operator to its chain recurrent set is topologically mixing and Devaney chaotic. We prove that topologically stable operators on locally convex spaces have the finite shadowing property and the strict periodic shadowing property. As a consequence, topologically stable operators on Banach spaces have the shadowing property. Moreover, we prove that generalized hyperbolicity implies topological stability for operators on Banach spaces. We prove that uniformly topologically expansive operators on locally convex spaces are neither Li-Yorke chaotic nor topologically transitive. Finally, we characterize the notion of topological expansivity for weighted shifts on Fréchet sequence spaces. Several examples are provided.

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”.

Connectedness in New Pseudo-Topological Vector Spaces

Studying connectedness is an important topic in classical topology, so we sought to study this property in the new pseudo topological vector space (T_(p ̃VS)) which is built on the convergence of the filters.

Theorem of Alternative and Scalarization of Optimization Problems in Topological Vector Spaces

In this paper, we introduce the definitions of affinelike, preaffinelike, generalized affinelike, and generalized preaffinelike functions by use of "pointed convex cone", and prove that those definitions of generalized affine functions are all different. We discuss optimization problems in topological vector spaces and obtain a theorem of alternative and a scalarization theorem. Our inequalities are given by partial order relations. Our generalized affineness may be used for many discussions in mathematics or applied mathematics wherever the affineness is a condition.

Multiplier rules for Dini-derivatives in a topological vector space

We provide new results of first-order necessary conditions of optimality problem in the form of John's theorem and in the form of Karush–Kuhn–Tucker's theorem. We establish our result in a topological vector space for problems with inequality constraints and in a Banach space for problems with equality and inequality constraints. Our contributions consist in the extension of the results known for the Fréchet and Gateaux-differentiable functions as well as for the Clarke's subdifferential of Lipschitz functions to the more general Dini-differentiable functions. As consequences, we extend the result of B.H. Pourciau in [Modern multiplier rules. Amer Math Monthly. 1980;87(6):433–457, Theorem 6, p. 445] from the convexity to the Dini-pseudoconvexity.

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.

A Really Simple Elementary Proof of the Uniform Boundedness Principle in F -Spaces

Abstract We give a proof of the uniform boundedness principle for linear continuous maps from F -spaces into topological vector spaces which is elementary and also quite simple.

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.

On the running and the UV limit of Wilsonian renormalization group flows

In nonperturbative formulation of quantum field theory, the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman type field correlators. Such a flow is a parametric family of ultraviolet (UV) regularized field correlators, the parameter being the strength of the UV regularization, and the instances with different strength of UV regularizations are linked by the renormalization group equation. Important RG flows are those which reach out to any UV regularization strengths. In this paper it is shown that for these flows a natural, mathematically rigorous generally covariant definition can be given, and that they form a topological vector space which is Hausdorff, locally convex, complete, nuclear, semi-Montel, Schwartz. That is, they form a generalized function space having favorable properties, similar to multivariate distributions. The other theorem proved in the paper is that for Wilsonian RG flows reaching out to all UV regularization strengths, a simple factorization formula holds in case of bosonic fields over flat (affine) spacetime: the flow always originates from a regularization-independent distributional correlator, and its running satisfies an algebraic ansatz. The conjecture is that this factorization theorem should generically hold, which is worth future investigations.

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.

Coercive and Noncoercive Mixed Generalized Complementarity Problems

Impressed with the very recent developments of noncoercive complementarity problems and the use of recession sets in complementarity problems, here, we discuss mixed generalized complementarity problems in Hausdorff topological vector spaces. We used the Tikhonov regularization procedure, as well as arguments from the recession analysis, to establish the existence of solutions for mixed generalized complementarity problems without coercivity assumptions in Banach spaces.

Representations of Solutions of Time-Fractional Multi-Order Systems of Differential-Operator Equations

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties.

A note on closedness of convex hull of sets

The closedness of certain sets plays an important role in mathematics, especially in convex and functional analysis. In this short note, we give some connections between closedness of Minkowski sum of sets, closedness of the convex hull of sets, and other properties of the topological vector space.

A comonadicity theorem for partial comodules

We show that the category of partial comodules over a Hopf algebra H is comonadic over Vectk and provide an explicit construction of this comonad using topological vector spaces. The case when H is finite-dimensional is treated in detail. A study of partial representations of linear algebraic groups is initiated; we show that a connected linear algebraic group does not admit partiality.

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

Quasitopological vector spaces

In this paper, motivated by a diffeological vector space with the D-topology, we introduce the concept of quasitopological vector space, which is a vector space with a topology satisfying the conditions that the vector addition is separately continuous in each variable, and the scalar multiplication is jointly continuous. The following results are proved: (1) Every quasitopological finite-dimensional vector space is a topological vector space. (2) For each infinite-dimensional vector space, we define a T1 non-Hausdorff topology on it such that it is a quasitopological vector space but is not a topological vector space.

Uncountably infinite algebraic genericity and spaceability for sequence spaces

Let X be a topological vector space of complex-valued sequences and Y be a subset of X. We provide conditions for (X∖Y)∪{0} to contain uncountably infinitely many linearly independent dense vector subspaces of X. We also provide conditions for (X∖Y)∪{0} to contain uncountably infinitely many linearly independent closed infinite-dimensional vector subspaces of X. We apply these results to a chain of spaces containing the ℓp spaces.