On additive correspondences

2015 ◽  
Vol 21 (1) ◽  
Author(s):  
Masoumeh Aghajani ◽  
Andrzej Smajdor
Keyword(s):  
Existence Theorem ◽  
Linear Space ◽  
Normed Space ◽  
Convex Cone ◽  
Normed Linear Space ◽  
Convex Compact ◽  
Open Convex ◽  
Novi Sad ◽  
Real Normed Linear Space

AbstractThe existence of additive selections of additive correspondences was investigated in [Ark. Mat. 4 (1960), 87–97], [Rev. Roumaine Math. Pures Appl. 28 (1983), 239–242.], [Math. Ser. Univ. Novi Sad 18 (1988), 143–148] and other papers. In this article, we find an existence theorem for additive selections of additive correspondences with convex compact values in a real normed linear space defined on an open convex cone of a real separable normed space.

scholarly journals On Uniqueness of New Orthogonality via 2-HH Norm in Normed Linear Space

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Bhuwan Prasad Ojha ◽  
Prakash Muni Bajracharya ◽  
Vishnu Narayan Mishra
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Inner Product ◽  
Isosceles Orthogonality ◽  
Complete Proof ◽  
Underlying Space ◽  
Real Inner Product Space ◽  
Special Case ◽  
Real Normed Linear Space ◽  
Pythagorean Orthogonality

This paper generalizes the special case of the Carlsson orthogonality in terms of the 2-HH norm in real normed linear space. Dragomir and Kikianty (2010) proved in their paper that the Pythagorean orthogonality is unique in any normed linear space, and isosceles orthogonality is unique if and only if the space is strictly convex. This paper deals with the complete proof of the uniqueness of the new orthogonality through the medium of the 2-HH norm. We also proved that the Birkhoff and Robert orthogonality via the 2-HH norm are equivalent, whenever the underlying space is a real inner-product space.

Linear Normed Spaces with Extension Property

1966 ◽  
Vol 9 (4) ◽  
pp. 433-441 ◽  
Author(s):  
George Elliott ◽  
Israel Halperin
Keyword(s):  
Linear Space ◽  
Linear Extension ◽  
Normed Linear Space ◽  
Extension Property ◽  
Linear Mapping ◽  
Normed Spaces ◽  
Bounded Linear ◽  
The Real ◽  
Real Normed Linear Space

In this paper we shall say “E has the (F, G) (extension) property” to mean the following: F is a subspace of the real normed linear space G, E is a real normed linear space, and any bounded linear mapping F→E has a linear extension G→E with the same bound (equivalently, every linear mapping F→E of bound 1 has a linear extension G→E with bound 1).

scholarly journals Fuzzyn-normed linear space

2005 ◽  
Vol 2005 (24) ◽  
pp. 3963-3977 ◽  
Author(s):  
AL. Narayanan ◽  
S. Vijayabalaji
Keyword(s):  
Linear Space ◽  
Normed Space ◽  
Best Approximation ◽  
Normed Linear Space

The primary purpose of this paper is to introduce the notion of fuzzyn-normed linear space as a generalization ofn-normed space. Ascending family ofα-n-norms corresponding to fuzzyn-norm is introduced. Best approximation sets inα-n-norms are defined. We also provide some results on best approximation sets inα-n-normed space.

scholarly journals Summable Family in a Commutative Group

2015 ◽  
Vol 23 (4) ◽  
pp. 279-288
Author(s):  
Roland Coghetto
Keyword(s):  
Topological Space ◽  
Linear Space ◽  
Metric Space ◽  
Normed Linear Space ◽  
Linear Topological Space ◽  
Commutative Group ◽  
Image Filter ◽  
Directed Set ◽  
Generic Theory ◽  
Real Normed Linear Space

Summary Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [22], [7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space [30], a real normed linear space [29] and a linear topological space [14] with the concept of the limit of an image filter [16]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commutative group (“additive notation” in [17]), using the notion of filters.

scholarly journals On nonconvex retracts in normed linear spaces

2016 ◽  
Vol 32 (2) ◽  
pp. 259-264
Author(s):  
GUOWEI ZHANG ◽  
◽  
PENGCHENG LI ◽  
Keyword(s):  
Linear Space ◽  
Convex Subset ◽  
Normed Linear Space ◽  
Continuous Mapping ◽  
Nonempty Closed Convex Subset ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Real Normed Linear Space

Let E be a real normed linear space. A subset X ⊂ E is called a retract of E if there exists a continuous mapping r : E → X, a retraction, satisfying r(x) = x, x ∈ X. It is well known that every nonempty closed convex subset of E is a retract of E. Nonconvex retracts are studied in this paper.

scholarly journals Contracting Mapping on Normed Linear Space

2012 ◽  
Vol 20 (4) ◽  
pp. 291-301
Author(s):  
Keiichi Miyajima ◽  
Artur Korniłowicz ◽  
Yasunari Shidama
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Space ◽  
Normed Space ◽  
Normed Linear Space ◽  
Contracting Mapping ◽  
Real Normed Space ◽  
The One

Summary In this article, we described the contracting mapping on normed linear space. Furthermore, we applied that mapping to ordinary differential equations on real normed space. Our method is based on the one presented by Schwarz [29].

scholarly journals A distance function property implying differentiability

1989 ◽  
Vol 39 (1) ◽  
pp. 59-70 ◽  
Author(s):  
J.R. Giles
Keyword(s):  
Linear Space ◽  
Distance Function ◽  
Normed Linear Space ◽  
Closed Set ◽  
Differentiable Norm ◽  
Function Property ◽  
Fréchet Differentiable ◽  
Image Position ◽  
Real Normed Linear Space

In a real normed linear space X, properties of a non-empty closed set K are closely related to those of the distance function d which it generates. If X has a uniformly Gâteaux (uniformly Fréchet) differentiable norm, then d is Gâteaux (Fréchet) differentiable at x ∈ X/K if there exists an such thatand is Géteaux (Fréchet) differentiable on X / K if there exists a set P+(K) dense in X/K where such a limit is approached uniformly for all x ∈ P+(K). When X is complete this last property implies that K is convex.

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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