Theory Of Topological Vector Spaces Research Articles

On integration with respect to filter

This article is devoted to the study of one generalization of the Riemann integral. Namely, in the paper, it was observed that the classical definition of the Riemann integral over a finite segment as a limit of integral sums, when the diameter of the division of the segment tends to zero, can be replaced by a limit of integral sums over a filter of sets, which can be described in a certain "good way". This idea was continued, and in the work we propose a new concept - the integral of a function over a filter on the set of all tagged partitions of a segment. Using of filters is a very good method in questions related to convergence or some of its analogues in general topological vector spaces. Namely, if the space is non-metrizable, then the concept of convergence is introduced precisely with the help of filters. Also, using filters, you can formulate the concept of completeness and its analogues. The completeness of spaces is one of the central concepts of the theory of topological vector spaces, since Banach spaces are complete. That is, using a generalization of the completeness of spaces constructed using filters, we can explore various generalizations of Banach spaces. We study standard issues related to integration. For example, does the integrability of the filter function imply its boundedness? The answer to this question is affirmative. Namely: the concept of filter boundedness of a function is introduced, and it is shown that if a function is integrable over filter, then its integral sums are bounded over the filter, and this function itself is bounded in the classical sense. Next, we showed that the filter integral satisfies the linearity property, namely, the integral over filter of the sum of two functions is the sum of the filter integrals of these functions. In addition, we can to subtract the constant factor from the sign of the integral over filter. We introduce the concept of an exactly tagged filter, and with the help of such filters we study the filter integrability of unbounded functions on a segment. We give an example of a specific unbounded function and a specific filter under which this function is integrable. Next, we prove a theorem that describes unbounded filter-integrable functions on a segment. The last section of the article is devoted to the integration of functions relative to the filter on a subsegment of this segment.

OLEG GEORGIYEVICH SMOLYANOV

The article is dedicated to the outstanding mathematician and pedagogue, Professor of Lomonosov Moscow State University, Oleg Georgiyevich Smolyanov (9 Feb. 1938 – 16 Dec. 2021), whose full life path and multifaceted activities were aimed at science and education development in the USSR and Russia. Oleg Georgiyevich, being a talented pedagogue and organiser, was a devotee of the scientific school in the field of the theory of topological vector spaces, infinite-dimensional analysis, beneficial for the formation of a huge number of scientists and teachers of higher educational institutions.

Regularity in Topological Modules

The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the literature of Functional Analysis with deeper classical results as well as with pathological phenomena. Following this trend, it has been recently proved that every real or complex Hausdorff locally convex topological vector space with dimension greater than or equal to 2 has a balanced and absorbing subset with empty interior. Here we propose an extension of this result to topological modules over topological rings. A sufficient condition is provided to accomplish this extension. This sufficient condition is a new property in topological module theory called strong open property. On the other hand, topological regularity of closed balls and open balls in real or complex normed spaces is a trivial fact. Sufficient conditions, related to the strong open property, are provided on seminormed modules over an absolutely semivalued ring for closed balls to be regular closed and open balls to be regular open. These sufficient conditions are in fact characterizations when the seminormed module is the absolutely semivalued ring. These characterizations allow the provision of more examples of closed-unit neighborhoods of zero. Consequently, the closed-unit ball of any unital real Banach algebra is proved to be a closed-unit zero-neighborhood. We finally transport all these results to topological modules over topological rings to obtain nontrivial regular closed and regular open neighborhoods of zero. In particular, if M is a topological R-module and m∗∈M∗ is a continuous linear functional on M which is open as a map between topological spaces, then m∗−1(int(B)) is regular open and m∗−1(B) is regular closed, for B any closed-unit zero-neighborhood in R.

Extreme points of a continuum

We import into continuum theory the notion of extreme point of a convex set from the theory of topological vector spaces. We explore how extreme points relate to other established types of “edge point” of a continuum; for example we prove that extreme points are always shore points, and that any extreme point is also non-block if the continuum is either decomposable or irreducible (in particular, metrizable).In addition we discuss some continuum-theoretic analogues of the celebrated Krein-Milman theorem.

$L^{2}$ Serre duality on domains in complex manifolds and applications

An L 2 version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the @-operator is established. This duality is used to study the solution of the @-equation with prescribed support. Applications are given to @-closed extension of forms, as well to Bochner-Hartogs type extension of CR functions. comp (;E � ) with the quotient topology, where we endow spaces of compactly supported forms with the natural inductive limit topology. In fact, condition that the two maps in (1) have closed range is also necessary for the duality theorem to hold (see (9); also see (26, 27, 28) for further results of this type.) Serre's original proof (33) is based on sheaf theory and the theory of topological vector spaces. A different approach to this result, in the case when is a compact complex manifold, was given by Kodaira using Hodge theory (see (23) or (7).) In this note we extend Kodaira's method to non-compact Hermitian manifolds to obtain an L 2 analog of the Serre duality. Special cases of Serre-duality using L 2 methods have appeared before in many contexts (see (25), or (11, Theorem 5.1.7) and (19, 20), for example.) Our treatment aims to streamline and systematize these results, with emphasis on non-compact manifolds, and point out its close relation with the choice of L 2 -realizations of the Cauchy-Riemann operator @, or alternatively, choice of boundary conditions for the L 2 -realizations of the formal complex Laplacian @E#E + #E@E. The L 2 -duality can be interpreted in many ways. At one level, it is a duality between the standard � - Laplacian with @-Neumann boundary conditions, and thec-Laplacian with dual ( @-Dirichlet) boundary conditions. Using another approach, results regarding solution of the @-equation in L 2 can be converted to statements regarding the solution of the @c equation. This leads to a solution of the @-Cauchy problem, i.e., solution of the @-equation with prescribed support. At the heart of the matter lies the existence of

Solutions of D. A. Raikov’s problems in the theory of topological vector spaces

Solutions of D. A. Raikov’s problems in the theory of topological vector spaces

Pontryagin Duality in the Theory of Topological Vector Spaces and in Topological Algebra

The theory of topological vector spaces (TVS), being a foundation of modern functional analysis, is now considered as a completely mature, or, to be more specific, dead mathematical discipline. This pessimistic view is based on a picture, where, on the one hand, a well-known system of facts is stated, facts that have been considered classical since the times of Mackey and Grothendieck, and, on the other hand, an opinion exists explicitly or implicitly, that any deviation from this system inevitably tends to a disappointment. “Do not seek any other patterns or connections but those that we have described here, because any of your assumptions will find a counterexample in Nature”; this phrase should have been cited as an afterward to the three textbooks on topological vector spaces translated into Russian [12, 41, 43], which represent the face of this science from the sixties until now. Indeed, after the famous Enflo counterexample [22], the development of the theory of topological vector spaces (which was planned originally as an extension of a clear and simple discipline, linear algebra) was transformed into a sequence of reports on the oddities one can encounter in modern mathematics. The result of this process is a general skepticism about and a loss of interest in this field. The number of papers on the theory of TVS is declining, while the attitude of other mathematicians now consists of polite perplexity or lenient irony. And add to that the fact that the vast majority of specialists in functional analysis do not concern themselves with topological vector spaces but consciously concentrate on Banach theory. (According to Compumath, for example, in 1996 only 7 papers were devoted to locally convex and topological vector spaces, while 612 were devoted to Banach spaces. In the theory of topological algebras, the ratio is 6:90.) The dominant position of Banach theory in functional analysis justifies a natural question: what are the advantages of the Banach case over the more general topological situation? At the conceptual level, the difference becomes most apparent in the theory of topological algebras. Let us cast a critical look. The first impression we will get when comparing Banach and non-Banach theories of topological algebras is as follows. While the concept of Banach algebra emerges naturally from intuitive expectations, faciliated by an acquaintance with general algebra and the theory of Banach spaces (and leads to the profound theory of Banach algebras), the situation with general topological algebras looks completely different. We find here unexpectedly that the very attempts to define a topological algebra (and topological module) lead to results that contradict intuition. Indeed, intuitively it is clear that a “good” definition of topological algebra (and topological module) must fulfill at least the following minimal list of requirements:

Some properties of Banach-valued sequence spacesℓp[X]

We discuss some properties of the Banach-valued sequence spaceℓp[X](1≤p<∞), the space of weaklyp-summable sequences on a Banach spaceX. For example, we characterize the reflexivity ofℓp[X], convergent sequences onℓp[X], and compact subsets ofℓp[X].

On Mackey topology for groups

The present paper is a contribution to fill in a gap existing between the theory of topological vector spaces and that of topological abelian groups. Topological vector spaces have been extensively studied as part of Functional Analysis. It is natural to expect that some important and elegant theorems about topological vector spaces may have analogous versions for abelian topological groups. The main obstruction to get such versions is probably the lack of the notion of convexity in the framework of groups. However, the introduction of quasi-convex sets and locally quasi-convex groups by Vilenkin [26] and the work of Banaszczyk [1] have paved the way to obtain theorems of this nature. We study here the group topologies compatible with a given duality. We have obtained, among others, the following result: for a complete metrizable topological abelian group, there always exists a finest locally quasi-convex topology with the same set of continuous characters as the original topology. We also give a description of this topology as an G-topology and we prove that, for the additive group of a complete metrizable topological vector space, it coincides with the ordinary Mackey topology.

Pontryagin duality in the theory of topological vector spaces

Pontryagin duality in the theory of topological vector spaces

The dimension of Euclidean subspaces of quasi-normed spaces

The purpose of this article is to extend certain results which are known to hold for convex bodies to a class of non-convex bodies occurring in the theory of topological vector spaces. In the first section after this introduction an analogue of F. John's Theorem on the distance of a finite dimensional space from Euclidean space is obtained, and the result is shown to be best possible. The Dvoretzky-Rogers Lemma on the points of contact of a symmetric convex body with the ellipsoid of maximum volume contained within it is discussed for certain non-convex bodies. In the next part the Dvoretzky Theorem on the existence of ellipsoidal sections is shown to hold with the best possible estimate for the dimension of the sections. It follows from estimates involving cotype constants that the finite dimensional subspaces of Lp (0 < p < 1) possess large almost Hilbertian subspaces. The final section extends the theorem of S. Szarek relating the volume of a body to the existence of ellipsoidal sections.

Erratum to "The Nonstandard Theory of Topological Vector Spaces"

Erratum to "The Nonstandard Theory of Topological Vector Spaces"

Erratum to “The nonstandard theory of topological vector spaces” (Trans. Amer. Math. Soc. 172 (1972), 405–435)

Erratum to “The nonstandard theory of topological vector spaces” (Trans. Amer. Math. Soc. 172 (1972), 405–435)

A Proof of a Theorem by Dolezal

Recently, Dolezal [1] stated a theorem: Suppose$\{ {\varphi _n } \}$is a sequence in$\mathcal{D}$converging to zero with respect to the weak topology generated on$\mathcal{D}$by the subspace of the distributions of finite order in$\mathcal{D}'$. Then, $\{ {\varphi _n } \} \to 0$, for the initial topology of$\mathcal{D}$. In this paper a proof of the theorem based on the theory of topological vector spaces is presented.

The nonstandard theory of topological vector spaces

The nonstandard theory of topological vector spaces

The Lattice of Subalgebras of a Boolean Algebra

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.

Analytic Distribution Kernels

Introduction. In his study of the structure of distribution kernels (centering around his celebrated kernel theorem), L. Schwartz [11; 13] has studied the subclasses of regular and very regular distribution kernels, which may be characterized roughly by the fact that they carry infinitely differentiable functions into infinitely differentiable functions. It is our object in the present paper to extend this study to kernels which are analytically regular, i.e., which, again roughly, carry analytic functions into analytic functions. The motivation for such a study is provided by the fundamental solutions of elliptic equations with analytic coefficients. One consequence of our results is that for such equations the standard theorems on the analyticity of regular solutions imply the analyticity of the fundamental solution. The principal tool in our investigation is the general theory of topological vector spaces (particularly in certain forms given to it by A. Grothendieck) and especially the theory of topological tensor products. In ?1.1, we summarize the definitions from the theory of topological vector spaces which we shall use. In ?1.2, we give a brief summary of the definitions and principal results of Grothendieck'stheory of topological tensor products. In ?I.3, we summarize the results of Schwartz's study of distribution kernels, at least insofar as they relate to the generalizations to be given to analytic kernels. ?II is devoted to an intensive consideration of the space Gt(K) of real analytic functions on a compact subset of Rn. ?III states and proves our principal results on analytic distribution kernels. The writer should like to thank Professors L. Nachbin and F. E. Browder for discussion, suggestions, and criticism. The results of the present paper constitute a portion of the writer's Doctoral Dissertation [1] at the University of Sao Paulo, prepared while in residence at the Institute for Pure and Applied Mathematics, Rio de Janeiro.

Analytic distribution kernels

Introduction. In his study of the structure of distribution kernels (centering around his celebrated kernel theorem), L. Schwartz [11; 13] has studied the subclasses of regular and very regular distribution kernels, which may be characterized roughly by the fact that they carry infinitely differentiable functions into infinitely differentiable functions. It is our object in the present paper to extend this study to kernels which are analytically regular, i.e., which, again roughly, carry analytic functions into analytic functions. The motivation for such a study is provided by the fundamental solutions of elliptic equations with analytic coefficients. One consequence of our results is that for such equations the standard theorems on the analyticity of regular solutions imply the analyticity of the fundamental solution. The principal tool in our investigation is the general theory of topological vector spaces (particularly in certain forms given to it by A. Grothendieck) and especially the theory of topological tensor products. In ?1.1, we summarize the definitions from the theory of topological vector spaces which we shall use. In ?1.2, we give a brief summary of the definitions and principal results of Grothendieck'stheory of topological tensor products. In ?I.3, we summarize the results of Schwartz's study of distribution kernels, at least insofar as they relate to the generalizations to be given to analytic kernels. ?II is devoted to an intensive consideration of the space Gt(K) of real analytic functions on a compact subset of Rn. ?III states and proves our principal results on analytic distribution kernels. The writer should like to thank Professors L. Nachbin and F. E. Browder for discussion, suggestions, and criticism. The results of the present paper constitute a portion of the writer's Doctoral Dissertation [1] at the University of Sao Paulo, prepared while in residence at the Institute for Pure and Applied Mathematics, Rio de Janeiro.

Topologies in vector lattices

This paper is concerned with topologies for vector lattices (in the sense of (1)) or spaces that satisfy the axioms K 1, 2, 3′ and 4 of (5) that are given by neighbourhoods. The notions of convergence introduced by Kantorovitch(5) and Birkhoff(1) do not in general lead to topologies which can be denned in terms of open sets, as, even for directed systems, the notion of closure derived from them is not such that the closure of every set is closed. In order to ensure this Kantorovitch introduces an axiom (his K6) which excludes even some Banach lattices. These notions of convergence are based more on the lattice aspect than the vector-space aspect of a vector lattice. In this paper the reverse is true, and it deals essentially with the application of the theory of topological vector spaces, as developed by von Neumann(9), Mackey (6, 7) and others, to vector lattices.