Infinite Dimensional Topological Vector Space Research Articles

The Tychonoff theorem and invariant pseudodistances

In this article we introduce a method of constructing functions with claimed properties by using the Tychonoff theorem. As an application of this method we show that the Caratheodory distance cD of convex domains D in a complex, locally convex, Hausdorff, and infinite-dimensional topological vector space is approximated by the Caratheodory distances cD∩Y in finite-dimensional linear subspaces Y. Originally this result is due to Dineen, Timoney, and Vigue who apply ultrafilters in their proof.

On a semigroup problem

If \begin{document}$ S $\end{document} is a semigroup in \begin{document}$ \mathbb{R}^n $\end{document} that is not separated by a linear functional, then it is known that the closure of \begin{document}$ S $\end{document} is a group. We investigate a similar statement in an infinite dimensional topological vector space \begin{document}$ X $\end{document} . We show that if \begin{document}$ X $\end{document} is an infinite dimensional Banach space, then there exists a semigroup \begin{document}$ S\subset X $\end{document} , not separated by the continuous functionals supported by the closed linear span of \begin{document}$ S $\end{document} , for which the closure of the semigroup is not a group. If \begin{document}$ X $\end{document} is an infinite dimensional Frechet space, then the closure of a semigroup that is not separated is always a group if and only if \begin{document}$ X $\end{document} is \begin{document}$ \mathbb{R}^{\omega} $\end{document} , the countably infinite direct product of lines. Other infinite dimensional topological vector spaces, such as \begin{document}$ \mathbb{R}^{\infty} $\end{document} , the countably infinite direct sum of lines, are discussed. The Semigroup Problem has applications to the study of certain dynamical systems, in particular for the construction of topologically transitive extensions of hyperbolic systems. Some examples are shown in the paper.

On measurability of real-valued functions in infinite-dimensional topological vector spaces

Abstract Some related questions concerning the measurability properties of real-valued functions with respect to a certain class of measures are discussed.

The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures

One of the goals of this article is to describe a setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure and of the concentration property on metric measured spaces, inspired from classical examples of means. In some cases, we get a linear extension of the limit at infinity. Then, the mean value on an infinite product is defined, first for cylindrical functions and secondly taking the uniform limit. Finally, the mean value for the heuristic Lebesgue measure on a separable infinite dimensional topological vector space (e.g., on a Hilbert space) is defined. This last object, which is not the classical infinite dimensional Lebesgue measure but its “normalized” version, is shown to be invariant under translation, scaling, and restriction.

Transformations of measures via their generalized densities

In this note we describe algorithms for obtaining formulae for transformations of measures on infinite dimensional topological vector spaces or manifolds, generated by transformations of the domains of the measures and by transformations of the range.

Separation problem for a family of Borel and Baire G-powers of shift measures on $ \mathbb{R} $

The separation problem for a family of Borel and Baire G-powers of shift measures on $ \mathbb{R} $ is studied for an arbitrary infinite additive group G by using the technique developed in [L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York (1974)], [ A. N. Shiryaev, Probability [in Russian], Nauka, Moscow (1980)], and [G. R. Pantsulaia, Invariant and Quasiinvariant Measures in Infinite-Dimensional Topological Vector Spaces, Nova Sci., New York, 2007]. It is proved that $ {T_n}:{{\mathbb{R}}^n}\to \mathbb{R},\;n\in \mathbb{N} $ , defined by $$ {T_n}\left( {{x_1},\ldots,{x_n}} \right)=-{F^{-1 }}\left( {{n^{-1 }}\#\left( {\left\{ {{x_1},\ldots,{x_n}} \right\}\cap \left( {-\infty; 0} \right]} \right)} \right) $$ for (x 1,…, x n ) ∈ $ {{\mathbb{R}}^n} $ is a consistent estimator of a useful signal θ in the one-dimensional linear stochastic model $$ {\xi_k}=\theta +{\varDelta_k},\quad k\in \mathbb{N}, $$ where #(∙) is a counting measure, ∆ k , k ∈ $ \mathbb{N} $ , is a sequence of independent identically distributed random variables on $ \mathbb{R} $ with a strictly increasing continuous distribution function F, and the expectation of ∆1 does not exist.

On a three-space property for Lindelöf $\Sigma$-spaces, (WCG)-spaces and the Sobczyk property

Corson's example shows that there exists a Banach space $E$ which is not weakly normal but $E$ contains a closed subspace isomorphic to the Banach space $C[0,1]$ and such that the quotient space $E/C[0,1]$ is isomorphic to the weakly compactly generated Banach space $c_{0}[0,1]$. This applies to show the following two results: (i) The Lindel\"of property is not a three-space property. (ii) The Lindel\"of $\Sigma$-property is not a three-space property. In this note using the lifting property developed by Susanne Dierolf we present a very simple argument providing also (ii), see Theorem 1. This argument used in the proof applies also to show that under Continuum Hypothesis every infinite-dimensional topological vector space $E$ which contains a dense hyperplane admits a stronger vector topology $\upsilon$ with the same topological dual and such that $(E,\upsilon)$ contains a dense non-Baire hyperplane. This partially answers a question of Saxon concerning Arias de Reyna-Valdivia-Saxon theorem. A Banach space $E$ has the Sobczyk Property if it contains an isomorphic copy of $c_0$ and every such a copy is complemented in $E$. The classical Sobczyk's theorem says that every separable Banach space has this property. We give an example of a $C(K)$-space $E$ and its subspace $Y$ isometric to $c_0$ such that $E/Y$ is isomorphic to $c_0(\Gamma)$, with $card(\Gamma )=2^{\aleph_0}$, yet $Y$ is uncomplemented in $E$. This complements Corson's example and shows that the Sobczyk Property (as well as the (WCG)-property, and the Separable Complementation Property) is not a~three-space property. In the last part we recall some facts (partially with a simpler presentation) concerning K-analytic, Lindelöf $\Sigma$ and analytic locally convex spaces. Additionally, a few remarks concerning weakly K-analytic spaces are included.

On realcompact topological vector spaces

This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vector spaces appearing in Functional Analysis, including Fréchet spaces, (LF)-spaces, and their duals, (DF)-spaces and spaces of continuous real-valued functions C(X) on a completely regular Hausdorff space X. Especially (LF)-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, Differential Equations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis.

Pareto equilibria for generalized constrained multiobjective games in FC-spaces without local convexity structure

In this paper, we further study a class of generalized constrained multiobjective games where the number of players may be finite or infinite, the strategy sets may be general FC-spaces without local convexity structure, and all payoff functions get their values in infinite-dimensional topological vector spaces. By using an existence theorem of maximal elements for a family of set-valued mappings in FC-spaces due to the author, an existence theorem of solutions for a system of generalized vector quasivariational inclusions is first proved in general FC-spaces. By applying the existence result of solutions of the system of generalized vector quasivariational inclusions, some existence theorems of (weak) Pareto equilibria for the generalized constrained multiobjective games are established in noncompact product FC-spaces. Some special cases of our results are also discussed. Our results are new and different from the corresponding known results in the literature.

On the Generic (Im)Possibility of Full Surplus Extraction in Mechanism Design

A number of studies, most notably Cremer and McLean (1985, 1988), have shown that in Harsanyi type spaces of a fixed finite size, it is generically possible to design mechanisms that extract all the surplus from players, and as a consequence, implement any outcome as if the players’ private information were commonly known. In contrast, we show that within the set of common priors on the universal type space, the subset of priors that permit the extraction of the players’ full surplus is shy. Shyness is an otion of smallness for convex subsets of infinite-dimensional topological vector spaces (in our case, the set of common priors), which generalizes the usual notion of zero Lebesgue measure in finite-dimensional spaces.

Comparison and Nuclearity of Spaces of Differential Forms on Topological Vector Spaces

Two types of fundamental spaces of differential forms on infinite dimensional topological vector spaces are considered; one is a fundamental space of Hida′s type and the other is one of Malliavin′s. It is proven that the former space is smaller than the latter. Moreover, it is shown that, under some conditions, the fundamental space of Hida′s type is nuclear as a complete countably normed space, while that of Malliavin′s in the L 2 sense is not.

Admissible vectors and their associated Dirichlet forms

Bouleau and Hirsch have introduced in [4] the energy-image density property and have proved that such property held for the Dirichlet form associated with the classical Wiener space. We will extend their result by proving that such energy-image density property holds more generally for the classical Dirichlet forms (see [1] for the terminology) on any infinite dimensional topological vector spaces provided with admissibility (introduced by Albeverio-Rockner in [1]).

Analytic Functionals on a Countably Infinite Dimensional Topological Vector Space

Analytic Functionals on a Countably Infinite Dimensional Topological Vector Space

Hypocontinuity of Multiplication on the Clifford Algebra of an Infinite-Dimensional Topological Vector Space

Hypocontinuity of Multiplication on the Clifford Algebra of an Infinite-Dimensional Topological Vector Space

Hypocontinuity of multiplication on the Clifford algebra of an infinite-dimensional topological vector space

Given a quadratic form on an infinite-dimensional vector space $E$, useful results have been obtained by imposing on $E$ the linear topology $t(V)$ described by Fischer and Gross [4], [5], [6], and investigated by Gross and Miller [9]. It has been shown that, in the induced topology, the Clifford algebra $C(E)$ is a topological algebra, but that, for topologies strictly finer than $t(V)$, multiplication need not be continuous. The main result of the present paper asserts that, even for topologies finer than $t(V)$, desirable conclusions can be drawn if continuity is replaced by hypocontinuity (see [2] for definition).

Locally-trivial bundles and microbundles with infinite-dimensional fibers

The object of this note is to establish trivialization theorems for locally-trivial bundles and microbundles in which the fiber is any one of a large class of infinite-dimensional topological vector spaces and the base space is any paracompact space. These theorems generalize results of Henderson and Wong.