scholarly journals On a Characterisation of Differentiability of the Norm of a Normed Linear Space

1971 ◽  
Vol 12 (1) ◽  
pp. 106-114 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Dual Norm ◽  
Uniform Rotundity ◽  
Strong Differentiability ◽  
Differentiability Properties

The purpose of this paper is to show that the various differentiability conditions for the norm of a normed linear space can be characterised by continuity conditions for a certain mapping from the space into its dual. Differentiability properties of the norm are often more easily handled using this characterisation and to demonstrate this we give somewhat more direct proofs of the reflexivity of a Banach space whose dual norm is strongly differentiable, and the duality of uniform rotundity and uniform strong differentiability of the norm for a Banach space.

scholarly journals A Novel Picture Fuzzy n-Banach Space with Some New Contractive Conditions and Their Fixed Point Results

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Awais Asif ◽  
Hassen Aydi ◽  
Muhammad Arshad ◽  
Zeeshan Ali
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Linear Space ◽  
Fuzzy Set ◽  
Fixed Point Theorems ◽  
Normed Linear Space ◽  
Real Life ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Picture Fuzzy Set

A picture fuzzy n-normed linear space (NPF), a mixture of a picture fuzzy set and an n-normed linear space, is a proficient concept to cope with uncertain and unpredictable real-life problems. The purpose of this manuscript is to present some novel contractive conditions based on NPF. By using these contractive conditions, we explore some fixed point theorems in a picture fuzzy n-Banach space (BPF). The discussed modified results are more general than those in the existing literature which are based on an intuitionistic fuzzy n-Banach space (BIF) and a fuzzy n-Banach space. To express the reliability and effectiveness of the main results, we present several examples to support our main theorems.

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 On a differentiability condition for reflexivity of a Banach space

1971 ◽  
Vol 12 (4) ◽  
pp. 393-396 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Differentiability Condition

In studying the geometry of normed linear space it is useful to draw attention to the following mapping.

scholarly journals Product-Normed Linear Spaces

2018 ◽  
Vol 11 (3) ◽  
pp. 740-750
Author(s):  
Benedict Barnes ◽  
I. A. Adjei ◽  
S. K. Amponsah ◽  
E. Harris
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Differential Equations ◽  
Linear Space ◽  
Linear Algebra ◽  
Functional Properties ◽  
Normed Linear Space ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Space Product

In this paper, both the product-normed linear space $P-NLS$ (product-Banach space) and product-semi-normed linear space (product-semi-Banch space) are introduced. These normed linear spaces are endowed with the first and second product inequalities, which have a lot of applications in linear algebra and differential equations. In addition, we showed that $P-NLS$ admits functional properties such as completeness, continuity and the fixed point.

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.

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