Topological Vector Spaces Research Articles

Denjoy-type Integrals in Locally Convex Topological Vector Space

In this paper, we introduce AC* and ACG*-type properties and then, using theseconditions along with other concepts, define two Denjoy-type integrals of a function with values in a locally convex topological vector space (LCTVS). We show, among others, that these newly defined integrals are included in the SH integral, a stronger version of the Henstock integral for LCTVS-valued functions.

A universal coregular countable second-countable space

A Hausdorff topological space X is called superconnected (resp. coregular) if for any nonempty open sets U1,…Un⊆X, the intersection of their closures U‾1∩…∩U‾n is not empty (resp. the complement X∖(U‾1∩…∩U‾n) is a regular topological space). A canonical example of a coregular superconnected space is the projective space QP∞ of the topological vector space Q<ω={(xn)n∈ω∈Qω:|{n∈ω:xn≠0}|<ω} over the field of rationals Q. The space QP∞ is the quotient space of Q<ω∖{0}ω by the equivalence relation x∼y iff Q⋅x=Q⋅y.We prove that every countable second-countable coregular space is homeomorphic to a subspace of QP∞, and a topological space X is homeomorphic to QP∞ if and only if X is countable, second-countable, and admits a decreasing sequence of closed sets (Xn)n∈ω such that (i) X0=X, ⋂n∈ωXn=∅, (ii) for every n∈ω and a nonempty relatively open set U⊆Xn the closure U‾ contains some set Xm, and (iii) for every n∈ω the complement X∖Xn is a regular topological space. Using this topological characterization of QP∞ we find topological copies of the space QP∞ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.

Extension of Monotonic Functions and Representation of Preferences

Consider a dominance relation (a preorder) ≿ on a topological space X, such as the greater than or equal to relation on a function space or a stochastic dominance relation on a space of probability measures. Given a compact set K ⊆ X, we study when a continuous real function on K that is strictly monotonic with respect to ≿ can be extended to X without violating the continuity and monotonicity conditions. We show that such extensions exist for translation invariant dominance relations on a large class of topological vector spaces. Translation invariance or a vector structure are no longer needed when X is locally compact and second countable. In decision theoretic exercises, our extension theorems help construct monotonic utility functions on the universal space X starting from compact subsets. To illustrate, we prove several representation theorems for revealed or exogenously given preferences that are monotonic with respect to a dominance relation.

ON THE SL-INTEGRAL OF LCTVS-VALUED FUNCTIONS

A function F:[a,b]→X is said to be an SL function if it satisfies the Strong Lusin (SL) condition given as follows: for every θ-nbd U and a set E⊂[a,b] of measure zero, there exists a gauge δ such that for every δ-fine partial partition D={([xi-1,xi],ti):1≤i≤n} of [a,b] with ti∈E, there exist θ-nbds U1,U2,…,Un such that ∑i=1nUi⊆V and F(xi)-F(xi-1)∈Ui for each i=1,2,…,n. In this paper, we introduce the SL integral of a function taking values on a locally convex topological vector space (LCTVS). Further, we show that this integral is equivalent to a stronger version of the Henstock integral.

On abstract economies with an arbitrary set of players and action sets in locally-convex topological vector spaces

In this paper we show the existence of equilibrium in an abstract economy with an arbitrary set of players, each of whose compact convex action sets lie in different locally convex spaces, and whose preferences obey weaker continuity postulates that have been used in the original treatments of Arrow and Debreu in the early fifties, and of Shafer and Sonnenschein in the mid-seventies. Our theorem synthesizes results around the cardinality of the set of agents, and incorporates in particular recent results of Carmona–Podczeck and He–Yannelis. The proof of the theorem is based on nonstandard analysis and its novelty has independent methodological interest.

Chaos on Fuzzy Dynamical Systems

Given a continuous map f:X→X on a metric space, it induces the maps f¯:K(X)→K(X), on the hyperspace of nonempty compact subspaces of X, and f^:F(X)→F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X→[0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f¯), and (F(X),f^). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).

An Intersection Theorem for Topological Vector Spaces and Applications

An Intersection Theorem for Topological Vector Spaces and Applications

Advances in the Theory of Compact Groups and Pro-Lie Groups in the Last Quarter Century

This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th centuries. Next is from Hilbert’s Fifth Problem in 1900 to its solution in 1952 by Montgomery, Zippin, and Gleason and Yamabe’s important structure theorem on almost connected locally compact groups. This half century included profound contributions by Weyl and Peter, Haar, Pontryagin, van Kampen, Weil, and Iwasawa. The focus in the last quarter century has been structure theory, largely resulting from extending Lie Theory to compact groups and then to pro-Lie groups, which are projective limits of finite-dimensional Lie groups. The category of pro-Lie groups is the smallest complete category containing Lie groups and includes all compact groups, locally compact abelian groups, and connected locally compact groups. Amongst the structure theorems is that each almost connected pro-Lie group G is homeomorphic to RI×C for a suitable set I and some compact subgroup C. Finally, there is a perfect generalization to compact groups G of the age-old natural duality of the group algebra R[G] of a finite group G to its representation algebra R(G,R), via the natural duality of the topological vector space RI to the vector space R(I), for any set I, thus opening a new approach to the Hochschild-Tannaka duality of compact groups.

Addendum to "$\alpha-$Topological Vector Spaces''[Hacet. J. Math. Stat. 47 (5), 1102-1107, 2018]

In this article, we give the corrected version of Theorem 3.5 and Theorem 3.7 (a) in [$\alpha-$Topological vector spaces, Hacet. J. Math. Stat. 47 (5), 1102-1107, 2018] by T. Al-Hawary.

Existence results for common solution of equilibrium and vector equilibrium problems

In this paper, by using the notion of locally segment-dense subsets and sequentially sign property for bifunctions, we establish existence results for a common solution of a finite family of equilibrium problems in the setting of Hausdorff locally convex topological vector spaces. Also similar resultsobtain for vector equilibrium problems.

Comment on Strongly Preirresolute Topological Vector Spaces

Let (X, =) be a topological space. A subset A of X is called&#x0D; pre-open if A ⊆ Int(Cl(A)). Let P O(X) denote the family of all pre-open&#x0D; sets in X. In general, P O(X) does not form a topology on X. Furthermore, in topological vector spaces, it is not always true that P O(L) forms&#x0D; a topology on L where L is a topological vector space. In this note, we&#x0D; prove that the class of strongly preirresolute topological vector spaces is&#x0D; that subclass of topological vector spaces in which P O(L) forms a topology and thereby we see that all proved results in [5] concerning strongly&#x0D; preirresolute topological vector spaces are obvious.

Generalized harmonic functions on trees: Universality and frequent universality

Recently, harmonic functions and frequently universal harmonic functions on a tree T have been studied, taking values on a separable Fréchet space E over the field C or R. In the present paper, we allow the functions to take values in a vector space E over a rather general field F. The metric of the separable topological vector space E is translation invariant and instead of harmonic functions we can also study more general functions defined by linear combinations with coefficients in F. We don't assume that E is complete and therefore we present an argument avoiding Baire's theorem.

Convex analysis of minimal time and signed minimal time functions

In this paper, we first consider the class of minimal time functions in the general setting of locally convex topological vector (LCTV) spaces. The results obtained in this framework are based on a novel notion of the closedness of target sets with respect to constant dynamics. Then, we introduce and investigate a new class of signed minimal time functions, which are generalizations of the signed distance functions. Subdifferential formulas for the signed minimal time and distance functions are obtained under the convexity assumptions on the given data.

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.

Existence of solutions for extended generalized complementarity problems

In this paper we introduce extended generalized complementarity problems in Hausdorff topological vector spaces in duality and study the existence of their solutions. We use a different method than those in literature on the existence of solutions of complementarity problems, which are usually based on arguments from generalized monotonicity. This leads us to obtain new results and improve many existing results in literature. We also prove some existence results for extended generalized complementarity problems in reflexive Banach spaces by means of a Tikhonov regularization procedure under a copositivity assumption and arguments from the recession analysis.

The rise and fall of L-spaces, II

In 2005, Ben-El-Mechaiekh, Chebbi, and Florenzano obtained a generalization of Ky Fan's 1984 KKM theorem on the intersection of a&#x0D; family of closed sets on non-compact convex sets in a topological vector space. They also extended the Fan-Browder fixed point theorem&#x0D; to multimaps on non-compact convex sets. Since then several groups of the L-space theorists introduced coercivity families and applied&#x0D; them to L-spaces, H-spaces, etc. In this article, we show that better forms of such works can be deduced from a general KKM theorem&#x0D; on abstract convex spaces in our previous works. Consequently, all of the known KKM theoretic results on L-spaces related coercivity&#x0D; families are extended to corresponding better forms on abstract convex spaces.&#x0D; &#x0D; This article is a continuation of our \cite{38} and a revised and extended version of \cite{34}.

A Class of Generalized Operator Quasi-Equilibrium Problems

In this work, introduce and study a generalized operator quasi-equilibrium problems (in short, OQEP) in the setting of topological vector spaces. We prove some new existence results for the solution of this problem by applying <I>C</I>(<i>f</i>)-quasiconvex, escaping sequence in Hausdörff topological vector spaces. The results of this paper can generalize and unify previously known corresponding results of this area.

Fréchet Spaces via Fuzzy Structures

The concept of fuzzy Fréchet space Z introduced in two equivalent ways, the first way when Z considered as a fuzzy F − locally convex topological vector space and the second one when Z defined as a complete fuzzy topological vector space whose fuzzy topology generated by a countable separating collection of fuzzy seminorms. Moreover, many classes of continuous mappings between these spaces are established and specified that the class of continuous, strongly continuous and sequentially continuous mappings between these spaces are equivalent.

Non-Linear Inner Structure of Topological Vector Spaces

Inner structure appeared in the literature of topological vector spaces as a tool to characterize the extremal structure of convex sets. For instance, in recent years, inner structure has been used to provide a solution to The Faceless Problem and to characterize the finest locally convex vector topology on a real vector space. This manuscript goes one step further by settling the bases for studying the inner structure of non-convex sets. In first place, we observe that the well behaviour of the extremal structure of convex sets with respect to the inner structure does not transport to non-convex sets in the following sense: it has been already proved that if a face of a convex set intersects the inner points, then the face is the whole convex set; however, in the non-convex setting, we find an example of a non-convex set with a proper extremal subset that intersects the inner points. On the opposite, we prove that if a extremal subset of a non-necessarily convex set intersects the affine internal points, then the extremal subset coincides with the whole set. On the other hand, it was proved in the inner structure literature that isomorphisms of vector spaces and translations preserve the sets of inner points and outer points. In this manuscript, we show that in general, affine maps and convex maps do not preserve inner points. Finally, by making use of the inner structure, we find a simple proof of the fact that a convex and absorbing set is a neighborhood of 0 in the finest locally convex vector topology. In fact, we show that in a convex set with internal points, the subset of its inner points coincides with the subset of its internal points, which also coincides with its interior with respect to the finest locally convex vector topology.

Tannaka Duality, Coclosed Categories and Reconstruction for Nonarchimedean Bialgebras

The topic of this paper is a generalization of Tannaka duality to coclosed categories. As an application we prove reconstruction theorems for coalgebras (bialgebras, Hopf algebras) in categories of topological vector spaces over a nonarchimedean field K. In particular, our results imply reconstruction and recognition theorems for categories of locally analytic representations of compact p-adic groups, which was the major motivation for this work.