Fuzzy bounded linear operators

The aim of this paper is to obtain several topological characteristics of fuzzy normed linear spaces. We study the vector topology induced by a fuzzy norm. The notion of Schauder basis is developed in fuzzy normed linear spaces.

Finite dimensional fuzzy normed linear space

In this paper we will highlight some properties of fuzzy bounded sets in fuzzy normed linear spaces. More precisely, we will prove that any finite union or sum of fuzzy bounded sets is fuzzy bounded. We show that the closure of a fuzzy bounded set is also fuzzy bounded. We give a characterization for fuzzy bounded sets and finally we obtain that the Cartesian product of two fuzzy bounded sets is also fuzzy bounded.

In this paper, we use the denition of fuzzy normed spaces given by Bag and Samanta and the behaviors of solutions of the additive functional equation are described. The Hyers-Ulam stability problem of this equation is discussed and theorems concerning the Hyers-Ulam-Rassias stability of the equation are proved on fuzzy normed linear space.

In this article we define the notion of fuzzy normed linear space valued sequence space lFp (X), 1 ≤ p < ∞ in a fuzzy normed linear space X and discuss some of its properties like completeness, monotone, solidness, convergence free and symmetricity. Also we prove some inclusion results.

This paper deals with few spectral properties for elements from a fuzzy normed algebra. It is introduced the definition of the spectral radius of an element from the fuzzy normed linear space (X, N, Λ) and some basic properties of it are shown. More precisely, we will prove that almost all the properties of this spectrum from some particular case of locally convex spaces are also true for this setup.

In this paper, we aim to present some properties of the space of all weakly fuzzy bounded linear operators, with the Bag and Samanta's operator norm on fuzzy normed linear spaces. We introduced the natural linear injection of a fuzzy normed linear space X in to its second dual spaceX".

There are excellent books on both functional analysis and summability. Most of them are very terse. In Functional Analysis and Summability, the author makes a sincere attempt for a gentle introduction of these topics to students. In the functional analysis component of the book, the Hahn–Banach theorem, Banach–Steinhaus theorem (or uniform boundedness principle), the open mapping theorem, the closed graph theorem, and the Riesz representation theorem are highlighted. In the summability component of the book, the Silverman–Toeplitz theorem, Schur’s theorem, the Steinhaus theorem, and the Steinhaus-type theorems are proved. The utility of functional analytic tools like the uniform boundedness principle to prove some results in summability theory is also pointed out. Features A gentle introduction of the topics to the students is attempted. Basic results of functional analysis and summability theory and their applications are highlighted. Many examples are provided in the text. Each chapter ends with useful exercises. This book will be useful to postgraduate students, pre-research level students, and research scholars in mathematics. Students of physics and engineering will also find this book useful since topics in the book also have applications in related areas.

The first four sections of this chapter form its core and include classical topics such as bounded linear transformations, the open mapping theorem, the closed graph theorem, the uniform boundedness principle, and the Hahn-Banach theorem. The chapter includes a good number of applications of the four fundamental theorems of functional analysis. Sections 6.5 and 6.6 provide a good account of the properties of the spectrum and adjoint operators on Banach spaces. They may be largely bypassed, since the treatment of the corresponding topics for operators on Hilbert spaces in chapter 7 is self-contained. The section on weak topologies is more advanced and may be omitted if a brief introduction is the goal. The chapter is enriched by such topics as the best polynomial approximation, the Hilbert cube, Gelfand’s theorem, Schauder bases, complemented subspaces, and the Banach-Alaoglu theorem.

In this study, first, lacunary convergence of double sequences is introduced in fuzzy normed spaces, and basic definitions and theorems about lacunary convergence for double sequences are given in fuzzy normed spaces. Then, we introduce the concept of lacunary ideal convergence of double sequences in fuzzy normed spaces, and the relation between lacunary convergence and lacunary ideal convergence is investigated for double sequences in fuzzy normed spaces. Finally, in fuzzy normed spaces, we give the concept of limit point and cluster point for double sequences, and the theorems related to these concepts are given.

Normed spaces are treated at length as well as techniques of Banach spaces for solving differential and integral equations. Classical results, such as the Open Mapping Theorem, the Closed Graph Theorem, the Hahn-Banach Theorem, and a few more, are given as well, establishing the core of the theory of bounded operators on Banach spaces. The chapter closes with a view towards generalizations of the theory in two directions, providing a glimpse of the theory of unbounded operators and of locally convex spaces.

In this central chapter, the fundamental elements of functional analysis are presented. Although topological vector spaces are considered, essentially for the use of weak topologies, the focus is on normed spaces. Normed spaces in duality or metric duality form a convenient framework. The main pillars of functional analysis are presented: separation properties, the uniform boundedness theorem, the open mapping theorem, the closed graph theorem…Some special properties such as reflexivity, separability, and uniform convexity are given some attention. An account of spectral theory for linear operators is presented and the case of compact operators is considered. The chapter ends with a presentation of regulated functions and functions of bounded variation. This enables us to dispose of an elementary integration theory that is often sufficient for simple purposes.

In this paper, we introduce the definition of weakly convergent sequence in fuzzy normed spaces. We investigate relationship between convergent sequence and weakly convergent sequence in fuzzy normed spaces. We also study the dual spaces of a standard fuzzy normed space and $01$-fuzzy normed space.

The space $${\cal H}{\cal K}$$ of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Frechet spaces $${\cal H}{\cal K}$$ (X). In this paper, on each Frechet space $${\cal H}{\cal K}$$ (X), an F-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $${\cal H}{\cal K}$$ (X) space. It is known that every control-convergent sequence in the $${\cal H}{\cal K}$$ space always belongs to a $${\cal H}{\cal K}$$ (X) space for some X. We illustrate how to apply results for Frechet spaces $${\cal H}{\cal K}$$ (X) to control-convergent sequences in the $${\cal H}{\cal K}$$ space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.

In this paper, we introduce a new type of convergence for a sequence of function, namely, λ-statistically convergent sequences of functions in fuzzy 2-normed space, which is a natural generalization of convergence in fuzzy 2-normed space. In particular, we introduce the concepts of uniform λ-statistical convergence and pointwise λ-statistical convergence in the topology induced by fuzzy 2-normed spaces, and give some basic properties of these concepts.