Normed Linear Spaces Research Articles

Trapezoid Orthogonality in Complex Normed Linear Spaces

Let Gp(x,y,z)=∥x+y+z∥p+∥z∥p−∥x+z∥p−∥y+z∥p be defined on a normed space X. The special case G2(x,y,z)=0,∀z∈X, where X is a real normed linear space, coincides with the trapezoid orthogonality (T-orthogonality), which was originally proposed by Alsina et al. in 1999. In this paper, for the case where X is a complex inner product space endowed with the inner product ⟨·,·⟩ and induced norm ∥·∥, it is proved that Sgn(G2(x,y,z))=Sgn(Re⟨x,y⟩),∀z∈X, and a geometric explanation for condition Re⟨x,y⟩=0 is provided. Furthermore, a condition G2(x,iy,z)=0,∀z∈X is added to extend the T-orthogonality to the general complex normed linear spaces. Based on some characterizations, the T-orthogonality is compared with several other well-known types of orthogonality. The fact that T-orthogonality implies Roberts orthogonality is also revealed.

On uniform convexity of linear metric spaces

UDC 517 The notion of uniform convexity was introduced and discussed in normed linear spaces by Clarkson [Trans. Amer. Math. Soc., 40, 396–414 (1936)]. Later, this idea attracted attention of other researchers who studied these spaces in depth and also introduced and analyzed in detail numerous other weaker forms of uniform convexity. Further, the idea of uniform convexity and its weaker forms was also extended to linear metric spaces. Over the years, the researchers investigated and analyzed the properties of linear metric spaces equipped either with uniform convexity or with its weaker forms, and studied their relationships, inheritance of these properties by quotient spaces, and the applications of these properties to many other domains of mathematics. In this article, we survey the available literature on these topics in linear metric spaces.

Exploring Bounded Linear Operators in Neutrosophic Normed Linear Spaces

Exploring Bounded Linear Operators in Neutrosophic Normed Linear Spaces

Smoothness and approximate smoothness in normed linear spaces and operator spaces

Smoothness and approximate smoothness in normed linear spaces and operator spaces

An Exploration of Hesitant Fuzzy Normed Linear Space

In this paper, the notion of hesitant fuzzy norm based on the Bag-Samanta’s Type Fuzzy Norm on linear space has been introduced. Further the concepts of ascending family of semi-norms, convergence and fuzzy continuous linear operators are studied in hesitant fuzzy normed linear space.

Exploring Hahn-Banach Theorem: Insights into Functional Analysis and Topology

This paper delves into the Hahn-Banach Theorem, elucidating its significance and applications in functional analysis and topology. It commences by highlighting foundational concepts in normed linear spaces and gradually builds towards a comprehensive exploration of the theorem’s multiple forms. By reconstructing the proof from basic principles, the paper demonstrates the theorem’s versatility in extending linear functionals from subspaces to universal sets, initially within real linear spaces and subsequently in complex scenarios. The proof employs techniques like sublinearity, seminorms, and Zorn’s Lemma to articulate the theorem’s efficacy in both real and complex linear spaces. This study also discusses the adaptation of the theorem in normed linear spaces, where the absence of an explicit dominating function presents unique challenges, and continuity replaces boundedness to establish the theorem’s claims. Through rigorous analysis, the paper confirms the theorem’s pivotal role in linking points and functions within normed spaces, thereby enhancing understanding of dual spaces and their topological implications. Additionally, the practical applications of the theorem in establishing the existence of linear functionals and exploring the relationship between points and functional norms in normed spaces are also detailed, underscoring the theorem’s enduring relevance in mathematical discourse.

BIFURCATION FROM INFINITY IN A FUZZY NORMED SPACE

The aim of the paper is the study of bifurcation phenomena in fuzzy normed linear space. We first define topological degrees (Leray-Schauder and Brouwer) and the index of an isolated zero in fuzzy linear normed space with topology induced by Felbin’s norm. Using this index, we prove the existence of the bifurcation of solutions from the 0-line and from infinity for functional equation defined by a compact mapping, we also give some global results.

Quasi statistically rough convergence of sequences in gradual normed linear spaces

In the present article, we set forth with the new notion of quasi statistically rough convergence in the gradual normed linear spaces. We establish significant results that present several fundamental properties of this new notion. We also introduce the notion of $st_{q}^{r}(\mathcal{G})$-limit set and prove that it is gradually closed, convex, and plays an important role for the quasi statistically boundedness of a sequence in a gradual normed linear space.

Умови екстремальності допустимого елемента для задачі відшу-кання узагальненого чебишовського центра кількох точок деякого полінормованого простору відносно множини цього простору

The problems related to the need to approximate complex mathematical objects in the best possible way with simpler and more convenient ones arise in various sections of mathematical science. An important class of approximation theory problems is the best simultaneous approximation of several elements. The problem of finding the Chebyshev center of several points of a linear normalized space relative to the set of this space can be attributed to the problems of best simultaneous approximation of several elements. This task consists in finding in a given set of linear normed space such a point (the relative Chebyshev center) the maximum distance to which from several fixed points of space would be the smallest, in other words not exceeding the maximum distance from the given points to any other point of this set. The problems of the best simultaneous approximation of several elements of a linear normed space by convex sets of this space from single positions were considered, in particular, in works [1, 2]. In practice, one has to deal with such problems, in which, when finding the Chebyshev's center of several given points of a linear normed space relative to the set of this space, appear weighted distances. The task of finding the weighted distances of the Chebyshev’s center was considered, in particular, in the paper [3]. In this work, the criteria of generalized Chebyshov’s center in the sense of the weighted distances of the of several points of a linear normed space relative to the convex set of this space, based on the duality ratio for the corresponding extremal problem, are established. If in the problem of the Chebyshev's center of several points of a linear normed space, in which the distances between points are determined by weighted norms, the weighted norms are replaced, generally speaking, by different norms given on the corresponding linear space, then we obtain the problem of the Chebyshev's center of several points of some polynormed space, which is considered in this work. It is clear that the problems about the Chebyshev's center of several points of a linear normed space, which were discussed above, are partial cases of the problem about the Chebyshev center of several points of some polynormed space.

Intuitionistic Type-2 Fuzzy Normed Linear Space and Some of Its Basic Properties

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

A dual mapping associated to a closed convex set and some subdifferential properties

In this paper, we establish some properties of the multivalued mapping (x, d) ⇒ DC (x; d) that associates to every element x of a linear normed space X the set of linear continuous functionals of norm d ≥ 0 and which separates the closed ball B (x; d) from a closed convex set C ⊂ X. Using this mapping we give links with other important concepts in convex analysis (ε-approximation element, ε-subdifferential of distance function, duality mapping, polar cone). Thus, we establish a dual characterization of ε-approximation elements with respect to a nonvoid closed convex set as a generalization of a known result of Garkavi. Also, we give some properties of univocity and monotonicity of mapping DC. Mathematics Subject Classification (2010): 32A70, 41A65, 46B20, 46N10. Received 18 May 2023; Accepted 27 November 2023

Relationship between Generalized Orthogonality and Gâteaux Derivative

This paper investigates the relationship between generalized orthogonality and Gâteaux derivative of the norm in a normed linear space. It is shown that the Gâteaux derivative of x in the y direction is zero when the norm is Gâteaux differentiable in the y direction at x and x and y satisfy certain generalized orthogonality conditions. A case where x and y are approximately orthogonal is also analyzed and the value range of the Gâteaux derivative in this case is given. Moreover, two concepts are introduced: the angle between vectors in normed linear space and the ⊥Δ coordinate system in a smooth Minkowski plane. Relevant examples are given at the end of the paper.

Level numbers of a bounded linear operator between normed linear spaces-II

We continue the study of level numbers and level vectors of a bounded linear operator between normed linear spaces. We prove that every linear operator on a finite-dimensional polyhedral Banach space possesses only finitely many level numbers. We also explore the level numbers of weighted permutation operators between ℓ p spaces. Furthermore, we introduce the concept of approximate level numbers and study its properties and relation with level numbers. In particular, our results indicate that the concept of level numbers can be potentially applied to the spectral theory of operators, in the setting of normed linear spaces.

On statistical convergence of order α of sequences in gradual normed linear spaces

Abstract In the current paper, we introduce the notion of statistical convergence of order α and strongly p-Cesàro summability of order α of sequences in the gradual normed linear spaces. We investigate several properties and a few inclusion relations of the newly introduced notions.

Mean p-angular distance orthogonality in normed linear spaces

In this paper, we introduce and study a new concept of orthogonality, namely, mean p-angular distance orthogonality in normed linear spaces. We investigate main properties of this type of orthogonality and its relation to some other previously defined generalized orthogonalities. Then, we study ?-existence, ?-diagonal existence and S-existence theorems for this type of orthogonality. Moreover, a new characterization of inner product spaces is given in terms of property (H), properly formulated for the mean p-angular distance orthogonality.

New results in gradual normed linear spaces in light of generalized statistical convergence via ideal

New results in gradual normed linear spaces in light of generalized statistical convergence via ideal

Deferred statistical convergence of double sequences in the context of gradual normed linear spaces

In this paper, we introduce the concepts of ?deferred statistical convergence? and ?strong deferred convergence? for double sequences in the setting of gradual normed linear spaces. Our study uncovers the differences between these two concepts. Additionally, we perform a thorough analysis of the mathematical properties associated with these notions and establish several implication relationships.

Neoteric Advancements in Analysis and Applied Mathematics

The theme of this special issue is “Neoteric Advancements in Analysis and Applied Mathematics”. Its goal is to look at new developments in fractional calculus, summability theory, fixed point theory, and special functions, as well as how they can be used to make the future more sustainable. This issue is mainly about discussing new ways to combine fractional integration formulas with special functions, using summability theory to study how things converge in neutrosophic normed linear spaces, creating new hybrid fixed point iterative processes, making fixed point theorems more general, and a lot of other important topics. By discussing these mathematical advancements, we hope to highlight their applicability and possible benefits to a variety of present-day challenges.

Triple sequences and deferred statistical convergence in the context of gradual normed linear spaces

In this article, we introduce the concepts of "deferred statistical convergence" and "strong deferred convergence" concerning triple sequences in the framework of gradual normed linear spaces. Our work reveals significant insights into the distinctions between these two concepts. Furthermore, we conduct an in-depth examination of the analytical properties of these notions and establish multiple implication relationships.

On Zweier J-convergent sequence spaces defined by Orlicz function in neutrosophic normed linear spaces

In this article we define and develop Zweier J-convergent sequence spaces: Z(M, R, B) (J : O) and Z0(M, R, B) (J : O) using Orlicz function in neutrosophic normed linear spaces. We also study some topological property of these spaces.