Topological Vector Spaces Research Articles

Trademark Empowerment using Optimal Neutrosophic Topological Vector Space for Maximizing Customer Attraction

Neutrosophic set is introduced as a generalization of intuitionistic fuzzy set, where any elements x ∈ X we have membership (T), non-membership (F), and indeterminacy (I)degrees. Neurosophic vague topological spaces are presented in various notations like neurosophic vague compactness and continuity. Trademarks are the essential components of intellectual property that allow owner to earn profit based on their name. In this industry, retailers typically use feedback channels like customer care service, website review complaints and suggestions boxes to gain user reviews on service satisfaction. But, there is a gap between these techniques. Customers are not fulfilled with them due to lack of trust in management, a lack of flexibility and slow responsiveness. This has prompted examination of the effect of customer feedback channels (CFCs) on client satisfaction and the necessity to develop a new CFC using artificial intelligence (AI). Thus, this study designs a Trademark Empowerment using Optimal Neutrosophic Topological Vector Space (TE-ONTVS) technique for Maximizing Customer Attraction. The intention of the TE-ONTVS technique lies in the prediction of customer behaviour and attraction. To accomplish this, the TE-ONTVS technique undergoes data scaling using Z-score normalization. In addition, the TE-ONTVS technique uses NTVS approach for the identification of customer behaviour and attraction. Lastly, whale optimization algorithm (WOA) is applied for optimal parameter tuning of the NTVS algorithm. A series of experiments were involved to demonstrate the enhanced outcomes of the TE-ONTVS algorithm. The obtained results stated that the TE-ONTVS technique reaches optimal performance over other models

Neutrosophic Topological Vector Spaces and its Properties

The algebraic structures Group, Ring, Field and Vector spaces are important innovations in Mathematics. Most of the theoretical concepts of Mathematics are based on the theorems related to these algebraic structures. Initially many mathematicians developed theorems related to all these algebraic structures. In 20th century most of the researchers introduced the theorems on the algebraic structures with Fuzzy and Intuitionistic fuzzy sets. Recently in 21st century the researchers concentrated on Neutrosophic sets and introduced the algebraic structures like Neutrosophic Group, Neutrosophic Ring, Neutrosophic Field, Neutrosophic Vector spaces and Neutrosophic Linear Transformation. In the current scenario of relating the spaces with the structures, we have introduced the concepts of Neutrosophic topological vector spaces. In this article, the study of Neutrosophic Topological vector spaces has been initiated. Some basic definitions and properties of classical vector spaces are generalized in Neutrosophic environment over a Neutrosophic field with continuous functions. Neutrosophic linear transformations and their properties are also included in Neutrosophic Topological Vector spaces. This article is an extension work of fuzzy and intuitionistic fuzzy vector spaces which were introduced in fuzzy and intuitionistic fuzzy environments. Even though it is an extension work, Neutrosophic Topological Vector space will play an important role in Neural Networks, Image Processing, Machine Learning and Artificial Intelligence Algorithms.

Exact Categories of Topological Vector Spaces with Linear Topology

Exact Categories of Topological Vector Spaces with Linear Topology

Schauder-Tychonoff Fixed Point Theorem on Sequentially Complete Hausdorff Strongly Convex Topological Vector Spaces

In this paper, we study the Schauder-Tychonoff fixed point (STFP) on a subset A of a sequentially complete Hausdorff strongly convex topological vector space (SCHSCTVS) E (over the field R) with calibration Γ have a unique STFP in Topological Vector Space (TVS).

CHARACTERISATION OF EQUIVALENT NORMS ON A LINEAR SPACE USING EXPONENTIAL VECTOR SPACE

In this paper we have found a necessary and sufficient condition for equivalence of two norms on a linear space using the theory of exponential vector space. Exponential vector space (‘evs’ in short) is an ordered algebraic structure which can be considered as an algebraic ordered extension of vector space. This structure is axiomatised on the basis of the intrinsic properties of the hyperspace C(X) comprising all nonempty compact subsets of a Hausdorff topological vector space X. Exponential vector space is a conglomeration of a semigroup structure, a scalar multiplication and a compatible partial order. We have shown that the collection of all norms defined on a linear space, together with the constant function zero, forms a topological exponential vector space. Then using the concept of comparing function (a concept defined on a topological exponential vector space) we have proved the aforesaid necessary and sufficient condition; also we have proved using comparing function that in an infinite dimensional linear space there are uncountably many non-equivalent norms.

TOPOLOGICAL MODULES GEOMETRY

The extremal structure of zero-neighbourhoods of a topological module is analyzed reaching unexpected conclusions when the module topology is not Hausdorff. These results motivate us to introduce the notion of metric modules, which are modules endowed with a translation-invariant metric, turning them into an (additive) topological group. We study the central and diametral points of additively symmetric subsets and find examples of convex sets which are not symmetric translates (translates of addivitely symmetric subsets). As a consequence of all of these, it seems natural to transport the well-known Bishop-Phelps property from the category of real topological vector spaces to general topological modules over topological rings. Then we stick to particular topological rings, the unital C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras, showing that the subset of positive elements lying below the unity is an effect algebra. We also prove that every continuous linear operator on a Hausdorff locally convex topological vector space that commutes with all continuous linear projections of one-dimensional range is a multiple of the identity. Finally, we discuss how to transport the previous result to C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras.

Topological semiinfinite tensor (super)modules

We construct universal monoidal categories of topological tensor supermodules over the Lie superalgebras gl(V⊕ΠV) and osp(V⊕ΠV) associated with a Tate space V. Here V⊕ΠV is a Z/2Z-graded topological vector space whose even and odd parts are isomorphic to V. We discuss the purely even case first, by introducing monoidal categories Tˆgl(V), Tˆo(V) and Tˆsp(V), and show that these categories are anti-equivalent to respective previously studied categories Tgl(V), To(V), Tsp(V). These latter categories have certain universality properties as monoidal categories, which consequently carry over to Tˆgl(V), Tˆo(V) and Tˆsp(V). Moreover, the categories To(V) and Tsp(V) are known to be equivalent, and this implies the equivalence of the categories Tˆo(V) and Tˆsp(V). After introducing a supersymmetric setting, we establish the equivalence of the category Tˆgl(V) with the category Tˆgl(V⊕ΠV), and the equivalence of both categories Tˆo(V) and Tˆsp(V) with Tˆosp(V⊕ΠV).

Stability of linear operators in locally convex cones

The theory of locally convex cones generalizes locally convex topological vector spaces and provides many different examples and applications than locally convex vector spaces. The stability problem in the sense of Ulam has not yet received any attention in the theory of locally convex cones. Therefore, in this paper, we introduce the stability problem in locally convex cones and provide the stability of linear operators in locally convex cones.

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.

New matching theorems for admissible type maps in topological vector spaces

New matching theorems for admissible type maps in topological vector spaces

Compact convex sets free of inner points in infinite‐dimensional topological vector spaces

AbstractAn inner point of a non‐singleton convex set is a point satisfying that for all there exists such that . We prove the existence of convex compact subsets free of inner points in the infinite‐dimensional setting. Following our pathway to this result, we come up with other several geometric hits, such as the existence of a non‐convex subset that coincides with its starlike envelope.

Multipliers of Variationally Mcshane Integrable Functions in Locally Convex Space

Abstract We study measurable real valued multipliers of variationally McShane (resp. McShane) integrable functions defined on a σ-finite outer regular quasi-Radon measure space and taking values in a complete locally convex topological vector space X. We also show: in case X is representable by semi-norm then essentially bounded real measurable functions are multipliers of functions which are Pettis integrable as well as integrable by semi-norm. The space of real valued measurable and essentially bounded functions turn out to be precisely the multipliers of variationally McShane (resp. McShane) integrable functions in case X is representable by semi-norm.

On Bishop–Phelps and Krein–Milman Properties

A real topological vector space is said to have the Krein–Milman property if every bounded, closed, convex subset has an extreme point. In the case of every bounded, closed, convex subset is the closed convex hull of its extreme points, then we say that the topological vector space satisfies the strong Krein–Milman property. The strong Krein–Milman property trivially implies the Krein–Milman property. We provide a sufficient condition for these two properties to be equivalent in the class of Hausdorff locally convex real topological vector spaces. This sufficient condition is the Bishop–Phelps property, which we introduce for real topological vector spaces by means of uniform convergence linear topologies. We study the inheritance of the Bishop–Phelps property. Nontrivial examples of topological vector spaces failing the Krein–Milman property are also given, providing us with necessary conditions to assure that the Krein–Milman property is satisfied. Finally, a sufficient condition to assure the Krein–Milman property is discussed.

Topological Vector Spaces Derived From Topological Hypervector Spaces

We introduce topological hypervector spaces on a topological field, in the sense of Tallini, and study some basic properties of this hyperspaces. In this regards we study the relationship between the topology on a hypervector spaces and its complete part. In particular we show that if every open subset of a topological hypervector space is a complete part then its fundamental vector space induced is a topological vector space. Finally, we study the quotient space of topological hypervector spaces and the derived topological space of a topological hypervector space with respect its fundamental relegation.

Once again on topological spaces containing a dense completely metrizable subset

We characterize the existence of a dense completely metrizable subset in a regular topological space X via the existence of residually defined continuous single-valued selections for certain set-valued mappings with values in X. We show that the property “X contains a dense completely metrizable subset” is preserved by continuous and demi-open single-valued mappings. We show also that the existence of residually defined continuous selections implies the Closed Graph and the Open Mapping theorems for completely metrizable topological vector spaces.

On semireflexive subcategories

In the category of Hausdorff locally convex topological vector spaces it is known that the class of L-semireflexive subcategories and some of its subclasses are isomorphic to classes of reflexive or coreflective subcategories. The most general case occurs when L is a c-reflective subcategory (L contains the subcategory of spaces with weak topology and the reflector functor is exactly to the left).This article examines a situation in which the coreflector functor commutes with products.

On some generalization of order cancellation law for subsets of topological vector space

Abstract In this paper we generalize a result of R. Urbański from paper [7] which states that for subsets A, B, C of topological vector space X the following implication holds A + B ⊂ B + C ⇒ A ⊂ C A + B \subset B + C \Rightarrow A \subset C provided that B is bounded and C is closed and convex. The generalization is given in Theorem 5 where we prove this result for k- convex subsets of a topological vector space. Also we introduce a notion of some abstract like-closure operation for subsets of linear space and we study its connections to order cancellation law.

Permutations, Signs, and Sum Ranges

The sum range SRx;X, for a sequence x=(xn)n∈N of elements of a topological vector space X, is defined as the set of all elements s∈X for which there exists a bijection (=permutation) π:N→N, such that the sequence of partial sums (∑k=1nxπ(k))n∈N converges to s. The sum range problem consists of describing the structure of the sum ranges for certain classes of sequences. We present a survey of the results related to the sum range problem in finite- and infinite-dimensional cases. First, we provide the basic terminology. Next, we devote attention to the one-dimensional case, i.e., to the Riemann–Dini theorem. Then, we deal with spaces where the sum ranges are closed affine for all sequences, and we include some counterexamples. Next, we present a complete exposition of all the known results for general spaces, where the sum ranges are closed affine for sequences satisfying some additional conditions. Finally, we formulate two open questions.

Fixed point theorems and applications in p-vector spaces

The goal of this paper is to develop new fixed points for quasi upper semicontinuous set-valued mappings and compact continuous (single-valued) mappings, and related applications for useful tools in nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p-convex and p-vector spaces for p in (0, 1]. In particular, we first develop general fixed point theorems for quasi upper semicontinuous set-valued and single-valued condensing mappings, which provide answers to the Schauder conjecture in the affirmative way under the setting of locally p-convex spaces and topological vector spaces for p in (0, 1]; then the best approximation results for quasi upper semicontinuous and 1-set contractive set-valued mappings are established, which are used as tools to establish some new fixed points for nonself quasi upper semicontinuous set-valued mappings with either inward or outward set conditions under various boundary situations. The results established in this paper unify or improve corresponding results in the existing literature for nonlinear analysis, and they would be regarded as the continuation of the related work by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022)–(Fixed Point Theory Algorithms Sci. Eng. 2022:26, 2022) recently.

A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces

This paper contains the equivalence between tvs-G cone metric and G-metric using a scalarization function $\zeta_p$, defined over a locally convex Hausdorff topological vector space. This function ensures that most studies on the existence and uniqueness of fixed-point theorems on G-metric space and tvs-G cone metric spaces are equivalent. We prove the equivalence between the vector-valued version and scalar-valued version of the fixed-point theorems of those spaces. Moreover, we present that if a real Banach space is considered instead of a locally convex Hausdorff space, then the theorems of this article extend some results of G-cone metric spaces and ensure the correspondence between any G-cone metric space and the G-metric space.