scholarly journals Complete Fuzzy n-normed linear space

2014 ◽  
Vol 3 (1) ◽  
Author(s):  
S. Vijayabalaji ◽  
N. Thillaigovindan
Keyword(s):  
Linear Space ◽  
Cauchy Sequence ◽  
Normed Linear Space ◽  
Convergent Sequence ◽  
Subject Classification

This paper introduces the notion of Cauchy sequence, convergent sequence and completeness in fuzzy n-normed linear space. AMS Mathematics Subject Classification: 46S40, 03E72.

Matrix Transformations in an Incomplete Space

1968 ◽  
Vol 20 ◽  
pp. 727-734 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Cauchy Sequence ◽  
Convergent Sequence ◽  
Convergent Series ◽  
Matrix Transformations ◽  
Cauchy Sequences ◽  
Complex Linear ◽  
Incomplete Space ◽  
Bounded Sequences ◽  
Convergent Sequences

Let X = (X, p) be a seminormed complex linear space with zero θ. Natural definitions of convergent sequence, Cauchy sequence, absolutely convergent series, etc., can be given in terms of the seminorm p. Let us write C = C(X) for the set of all convergent sequences for the set of Cauchy sequences; and L∞ for the set of all bounded sequences.

scholarly journals Interval valued Fuzzy n-normed linear space

2014 ◽  
Vol 4 (1) ◽  
Author(s):  
S. Vijayabalajia ◽  
S.Anita Shanthib ◽  
N. Thillaigovindanb
Keyword(s):  
Tensor Product ◽  
Linear Space ◽  
Normed Linear Space ◽  
Subject Classification ◽  
Ams Subject Classification ◽  
Interval Valued

The aim of this paper is to introduce the notion of interval-valued fuzzy subspace with its flags and interval-valued fuzzy n-normed linear space. We define the operations intersection, sum, directsum and tensor product of intervalvalued fuzzy subspaces and obtain their corresponding flags. Further we provide some results on interval-valued fuzzy n-normed linear space. AMS Subject Classification : 46S40, 03E72.

scholarly journals SOME RESULTS RELATED WITH FUZZY a −NORMED LINEAR SPACE

2019 ◽  
pp. 11-12
Author(s):  
M. Arunmaran ◽  
K. Kannan
Keyword(s):  
Linear Space ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Normed Linear Space ◽  
Finite Dimension ◽  
Convergent Sequence ◽  
Unit Interval ◽  
Clear Understanding ◽  
Convergent Sequences

Zadeh established the concept of fuzzy set based on the characteristic function. Foundation of fuzzy set theory was introduced by him. Throughout this paper, 𝑀𝑛(𝐹) denotes the set of all fuzzy matrices of order 𝑛 over the fuzzy unit interval [0,1]. Inaddi tion (𝑀𝑛 (𝐹), 𝜃) dis called as fuzzy 𝛼 −normed linear space. The objective of this paper is to investigate the relationships between convergent sequences and fuzzy 𝛼 −normed linear space. The set of all fuzzy points in 𝑀𝑛 (𝐹) is denoted by 𝑃∗(𝑀𝑛(𝐹)). For a fuzzy 𝛼 −normed linear space (𝑀𝑛 (𝐹), 𝜃), we have |𝜃(𝑃𝐴)𝛼 −𝜃(𝑃𝐵)𝛼 | ≤ 𝜃(𝑃𝐴,𝑃𝐵)𝛼. Besides 𝜃 is a continuous function on 𝑀𝑛 (𝐹). That is, if 𝑃𝐴𝑛 → 𝑃𝐴 as 𝑛 → ∞ then 𝜃(𝑃𝐴𝑛 )𝛼 → 𝜃(𝑃𝐴)𝛼 as 𝑛 → ∞, where 𝑃𝐴𝑛 is a sequence in (𝑀𝑛 (𝐹), 𝜃). Hence, 𝜃 is always bounded on 𝑀𝑛(𝐹). Next we introduce the following result: Let 𝑃𝐴𝑛 , 𝑃𝐵𝑛 ∈ 𝑃 ∗ (𝑀𝑛(𝐹)) with 𝑃𝐴𝑛 and 𝑃𝐵𝑛 converge to 𝑃𝐴 and 𝑃𝐵 respectively as 𝑛 → ∞. Then 𝑃𝐴𝑛 + 𝑃𝐵𝑛 converge to 𝑃𝐴 + 𝑃𝐵 as 𝑛 → ∞. Furthermore, we are able to compare two different fuzzy 𝛼 −norms with convergent sequence. The result states that for a fuzzy 𝛼 −normed linear space (𝑀𝑛(𝐹), 𝜃), we have 𝜃(𝑃𝐴)𝛼1 ≥ 𝑀𝜃(𝑃𝐴 )𝛼2 , for some 𝑀 > 0 and 𝑃𝐴 ∈ 𝑃 ∗ (𝑀𝑛(𝐹)). If 𝑃𝐴𝑛 converges to 𝑃𝐴 under fuzzy 𝛼1 −norm then 𝑃𝐴𝑛 converges to 𝑃𝐴 under fuzzy 𝛼2 −norm. Moreover, if (𝑀𝑛 (𝐹), 𝜃) has finite dimension then it should be complete. Through these results, we are able to get clear understanding about the concept fuzzy 𝛼 −normed linear space and its properties.

Weak Cauchy sequences in normed linear spaces

1964 ◽  
Vol 60 (4) ◽  
pp. 817-819 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Linear Space ◽  
Cauchy Sequence ◽  
Normed Linear Space ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Finite Dimensional ◽  
Cauchy Sequences ◽  
Convex Cover

It follows from the Krein-Milman theorem that (c0) is not isomorphic to the dual of a Banach space. Using a technique due to Banach ((4), page 194) we shall extend this result to show that if a subspace of (c0) is isomorphic to the dual of a normed linear space, then it is finite dimensional (Proposition 1). Using this result, we shall show that if E is a normed linear space, the unit ball of which is contained in the closed absolutely convex cover of a weak Cauchy sequence, then Eis finite dimensional (Proposition 2). This result has applications to the Banach-Dieudonné theorem, and to the theory of two-norm spaces.

scholarly journals Fixed points of quasi-nonexpansive mappings

1972 ◽  
Vol 13 (2) ◽  
pp. 167-170 ◽  
Author(s):  
W. G. Dotson
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Linear Space ◽  
Complex Plane ◽  
Normed Linear Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
The Other ◽  
Commuting Mappings ◽  
Analytic Mappings

A self-mapping T of a subset C of a normed linear space is said to be non-expansive provided ║Tx — Ty║ ≦ ║x – y║ holds for all x, y ∈ C. There has been a number of recent results on common fixed points of commutative families of nonexpansive mappings in Banach spaces, for example see DeMarr [6], Browder [3], and Belluce and Kirk [1], [2]. There have also been several recent results concerning common fixed points of two commuting mappings, one of which satisfies some condition like nonexpansiveness while the other is only continuous, for example see DeMarr [5], Jungck [8], Singh [11], [12], and Cano [4]. These results, with the exception of Cano's, have been confined to mappings from the reals to the reals. Some recent results on common fixed points of commuting analytic mappings in the complex plane have also been obtained, for example see Singh [13] and Shields [10].

scholarly journals Auerbach Bases and Minimal Volume Sufficient Enlargements

2011 ◽  
Vol 54 (4) ◽  
pp. 726-738
Author(s):  
M. I. Ostrovskii
Keyword(s):  
Linear Space ◽  
Isometric Embedding ◽  
Normed Linear Space ◽  
Convex Set ◽  
Linear Projection ◽  
Minimal Volume ◽  
Finite Dimensional ◽  
Dimensional Normed Space ◽  
Closed Convex Set

AbstractLet BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P: Y → X such that P(BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

Sign in / Sign up

Export Citation Format

Share Document