Some geometric and topological properties of the unit sphere in a normed linear space

1989 ◽  
pp. 339-344 ◽  
Author(s):  
Bor-Luh Lin ◽  
Pei-Kee Lin ◽  
S. L. Troyanski
Keyword(s):  
Linear Space ◽  
Unit Sphere ◽  
Normed Linear Space ◽  
Topological Properties

scholarly journals Suns are convex in tangent directions

2019 ◽  
Vol 484 (2) ◽  
pp. 131-133
Author(s):  
A. R. Alimov ◽  
E. V. Shchepin
Keyword(s):  
Linear Space ◽  
Unit Sphere ◽  
Normed Linear Space ◽  
Tangent Line ◽  
Chebyshev Set ◽  
Tangent Direction

A direction d is called a tangent direction to the unit sphere S of a normed linear space s  S and lin(s + d) is a tangent line to the sphere S at s imply that lin(s + d) is a one-sided tangent to the sphere S, i. e., it is the limit of secant lines at s. A set M is called convex with respect to a direction d if [x, y]  M whenever x, y in M, (y - x) || d. We show that in a normed linear space an arbitrary sun (in particular, a boundedly compact Chebyshev set) is convex with respect to any tangent direction of the unit sphere.

scholarly journals Multiplicative Normed Linear Space and its Topological Properties

2017 ◽  
Vol 10 (10) ◽  
pp. 1-9
Author(s):  
Renu Chugh ◽  
Ashish Nandal ◽  
Naresh Kumar ◽  
◽  
◽  
...  
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Topological Properties

scholarly journals The numerical range of an element of a normed algebra

1969 ◽  
Vol 10 (1) ◽  
pp. 68-72 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Linear Space ◽  
Unit Sphere ◽  
Dual Space ◽  
Normed Linear Space ◽  
Numerical Range ◽  
Normed Algebra ◽  
Bounded Linear ◽  
Image Position

Given a normed linear space X, let S(X), X′, B(X) denote respectively the unit sphere {x: ∥x∥ = 1} of X, the dual space of X, and the algebra of all bounded linear mappings of X into X. For each x ∊ S(X) and T ∊ B(X), let Dx(x) = {f e X′:∥f∥ = f(x)= 1}, and V(T; x) = {f(Tx):f∊Dx(x)}. The numerical range V(T) is then defined by

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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