Bornological Vector Spaces Research Articles

Mackey convergence and separation in $(L, M)$-fuzzy bornological vector spaces

This paper aims to introduce the concepts of Mackey convergence degree for sequences and separation degree for spaces in $(L, M)$-fuzzy bornological vector spaces. Additionally, the paper presents the concept of bornological closure degree for fuzzy sets. Moreover, the paper discusses various characteristics of these concepts. Furthermore, the paper examines the degree relationships among a Mackey convergence sequence, a separated space, and a bornologically closed fuzzy set. Finally, the paper analyzes the properties of functors $\omega$ and $\iota$ between $M$-fuzzifying bornological vector spaces and $(L, M)$-fuzzy bornological vector spaces in terms of Mackey convergence degree and separation degree.

Convergence of Filters on Bornological Vector Spaces and Neutrosophic Filters

In this research, we construct new type of convergence of bornological vector spaces called convergence of filters through using conception bounded sets. As well, we have considered several characteristics of these concepts like Fréchet filter associated with sequence, filter that has a unique limit and ultra-filter which is very useful in the study of neutrosophic topological spaces and neutrosophic filters.

(L,M)-fuzzy bornological spaces

In this paper, the notions of (L,M)-fuzzy bornological space and (L,M)-fuzzy bornological vector space are introduced. Some properties are discussed. It is shown that the category of (L,M)-fuzzy bornological (vector) spaces, which embeds the category of M-fuzzifying bornological (vector) spaces as a reflective subcategory, is topological category. Further, it is proved that the category of M-fuzzifying bornological spaces can be embedded in the category of stratified (L,M)-fuzzy bornological spaces as a coreflective subcategory.

Induced L-bornological vector spaces and L-Mackey convergence1

Motivated by the concept of lattice-bornological vector spaces of J. Paseka, S. Solovyov and M. Stehlík, which extends bornological vector spaces to the fuzzy setting over a complete lattice, this paper continues to study the theory of L-bornological vector spaces. The specific description of L-bornological vector spaces is presented, some properties of Lowen functors between the category of bornological vector spaces and the category of L-bornological vector spaces are discussed. In addition, the notions and some properties of L-Mackey convergence and separation in L-bornological vector spaces are showed. The equivalent characterization of separation in L-bornological vector spaces in terms of L-Mackey convergence is obtained in particular.

Orthogonally Additive Homogenous Polynomials on Vector Lattices

We study orthogonally additive homogeneous polynomials on a uniformly complete vector lattice taking values either in a separated bornological vector space or in a separated topological vector space. This study is based on the notions of orthosymmetric multilinear maps and vector lattice powers.

On the category of lattice-valued bornological vector spaces

Motivated by the theory of L-bornological spaces of M. Abel and A. Šostak over a complete lattice L, and the concept of topological system of S. Vickers, this paper introduces the categories of L-bornological vector spaces and systems, and shows that the former is isomorphic to a full reflective subcategory of the latter.

Caristi Type Coincidence Point Theorem in Topological Spaces

A generalized Caristi type coincidence point theorem and its equivalences in the setting of topological spaces by using a kind of nonmetric type function are obtained. These results are used to establish variational principle and its equivalences ind-complete spaces, bornological vector space, seven kinds of completed quasi-semimetric spaces equipped withQ-functions, uniform spaces withq-distance, generating spaces of quasimetric family, and fuzzy metric spaces.

Pontryagin duality for bornological quantum hypergroups

We develop a theory of bornological quantum hypergroups, aiming to extend the theory of algebraic quantum hypergroups in the sense of Delvaux and Van Daele to the framework of bornological vector spaces. It is very similar to the theory of bornological quantum groups established by Voigt, except that the coproduct is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We study the Fourier transform and develop Pontryagin duality theory for a bornological quantum hypergroup. As an application, we prove a formula relating the fourth power of the antipode with the modular functions of a bornological quantum hypergroup and its dual.

Pointwise bornological vector spaces

The existing duality between topological and bornological vector spaces allows us to define bornological objects in the category of topological vector spaces. For a Tychonoff space X and a set B of relatively pseudocompact subsets of X, the vector space C ( X ) endowed with the topology of uniform convergence on elements of B is a locally convex topological vector space, the bornological coreflection of which is described in [J. Schmets, Espaces de Fonctions Continues, Lecture Notes in Math., vol. 519, Springer-Verlag, Berlin, Heidelberg, New York, 1976; J. Schmets, Spaces of Vector-Valued Functions, Lecture Notes in Math., vol. 1003, Springer-Verlag, Berlin, Heidelberg, New York, 1983; J. Dontchev, S. Salbany, V. Valov, Barrelled and bornological function spaces, J. Math. Anal. Appl. 242 (2000) 1–17; J. Schmets, Spaces of vector-valued continuous functions, in: Proceedings Vector Space Measures and Applications I, Dublin, 1977, in: Lecture Notes in Math., vol. 644, Springer-Verlag, Berlin, Heidelberg, New York, 1978, pp. 368–377; J. Schmets, Bornological and ultrabornological C ( X , E ) spaces, Manuscripta Math. 21 (1977) 117–133]. If the elements of B are not supposed to be relatively pseudocompact, then this topology is no longer a vector topology and the bounded sets do not form a bornology, so the classical theory on bornologicity cannot be applied to it. The aim of this paper is to extend the duality between topological and bornological vector spaces to larger classes of objects and, moreover, apply it to C ( X ) endowed with three different, natural topologies.

Bornological quantum groups

We introduce and study the concept of a bornological quantum group. This generalizes the theory of algebraic quantum groups in the sense of van Daele from the algebraic setting to the framework of bornological vector spaces. Working with bornological vector spaces, the scope of the latter theory can be extended considerably. In particular, the bornological theory covers smooth convolution algebras of arbitrary locally compact groups and their duals. Moreover Schwartz algebras of nilpotent Lie groups are bornological quantum groups in a natural way, and similarly one may consider algebras of functions on finitely generated discrete groups defined by various decay conditions. Another source of examples arises from deformation quantization in the sense of Rieffel. Apart from describing these examples we obtain some general results on bornological quantum groups. In particular, we construct the dual of a bornological quantum group and prove the Pontrjagin duality theorem.

A drop theorem without vector topology

Daneš' drop theorem is extended to bornological vector spaces. An immediate application is to establish Ekeland-type variational principle and its equivalence, Caristi fixed point theorem, in bornological vector spaces. Meanwhile, since every locally convex space becomes a convex bornological vector space when equipped with the canonical von Neumann bornology, Qiu's generalization of Daneš' work to locally convex spaces is recovered.

Smooth group representations on bornological vector spaces

We develop the basic theory of smooth representations of locally compact groups on bornological vector spaces. In this setup, we are able to formulate better general theorems than in the topological case. Nonetheless, smooth representations of totally disconnected groups on vector spaces and of Lie groups on Fréchet spaces remain special cases of our theory. We identify smooth representations with essential modules over an appropriate convolution algebra. We examine smoothening functors on representations and modules and show that they agree if they are both defined. We establish the basic properties of induction and compact induction functors using adjoint functor techniques. We describe the center of the category of smooth representations.

A study on bornological properties of the space of entire functions of several complex variables

Spaces of entire functions of several complex variables occupy an important position in view of their vast applications in various branches of mathematics, for instance, the classical analysis, theory of approximation, theory of
 topological bases etc. With an idea of correlating entire functions with certain aspects in the theory of basis in locally convex spaces, we have investigated in this paper the bornological aspects of the space $X$ of integral functions of several complex variables. By $Y$ we denote the space of all power series with positive radius of convergence at the origin. We introduce bornologies on $X$
 and $Y$ and prove that $Y$ is a convex bornological vector space which is the completion of the convex bornological vector space $X$.

The uniform boundedness principle for bornological vector spaces

The uniform boundedness principle for bornological vector spaces

Bornologiese pseudotopologiese vektorruimtes

Homological spaces were defined by Hogbe-Nlend in 1971 and pseudo-topological spaces by Fischer in 1959. In this paper properties of bornological pseudo-topological vector spaces are investigated. A characterization of such spaces is obtained and it is shown that quotient spaces and direct sums o f boruological pseudo-topological vector spaces are bornological. Every bornological locally convex pseudo-topological vector space is shown to be the inductive limit in the category of pseudo-topological vector spaces of a family of locally convex topological vector spaces.

Structure of spaces ofC ∞-functions on nuclear spaces

LetE be a real nuclear locally convex space; we prove that the space ℰub(E), of allC ∞-functions of uniform bounded type onE, coincides with the inductive limit of the spaces ℰNbc(E v) (introduced by Nachbin-Dineen), whenV ranges over a basis of convex balanced 0-neighbourhoods inE. LetE be a real nuclear bornological vector space; we prove that the space ℰ(E) of allC ∞-functions onE coincides with the projective limit of the spaces ℰNbc(E B), whenB is a closed convex balanced bounded subset ofE. As a consequence we obtain some density results and a version of the Paley-Wiener-Schwartz theorem.

Convolution equations in spaces of infinite-dimensional entire functions of exponential and related types

We prove results of existence and approximation of the solutions of the convolution equations in spaces of entire functions of exponential type on infinite dimensional spaces. In particular we obtain: let E be a complex, quasi-complete and dual nuclear locally convex space and Ω \Omega a convex balanced open subset of E; let H ( Ω ) \mathcal {H} (\Omega ) be the space of the holomorphic functions on Ω \Omega , equipped with the compact open topology and H ′ ( Ω ) \mathcal {H}’(\Omega ) its strong dual; let F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) denote the image of H ′ ( Ω ) \mathcal {H}’(\Omega ) through the Fourier-Borel transform F \mathcal {F} ; equip this space F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) with the image of the topology of H ′ ( Ω ) \mathcal {H}’(\Omega ) via the map F \mathcal {F} . Then, “every nonzero convolution operator on F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) is surjective” and “every solution of the homogeneous equation is limit of exponential-polynomial solutions". Our results are more generally valid when E is a Schwartz bornological vector space with the approximation property. Previous results in Fréchet-Schwartz and Silva spaces are thus extended to domains that are not Fréchet or D.F.-spaces.