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1
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On the frame of the unit ball of Banach spaces

100%
Tanaka R.
Open Mathematics
|
2014
|
tom 12
|
nr 11
1700-1713
EN
The notion of the frame of the unit ball of Banach spaces was introduced to construct a new calculation method for the Dunkl-Williams constant. In this paper, we characterize the frame of the unit ball by using k-extreme points and extreme points of the unit ball of two-dimensional subspaces. Furthermore, we show that the frame of the unit ball is always closed, and is connected if the dimension of the space is not less than three. As infinite dimensional examples, the frame of the unit balls of c 0 and ℓ p are determined.
2
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On equivalent strictly G-convex renormings of Banach spaces

100%
Boyko N.
Open Mathematics
|
2010
|
tom 8
|
nr 5
871-877
EN
We study strictly G-convex renormings and extensions of strictly G-convex norms on Banach spaces. We prove that ℓω(Γ) space cannot be strictly G-convex renormed given Γ is uncountable and G is bounded and separable.
3
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On the calculation of the Dunkl-Williams constant of normed linear spaces

100%
Mizuguchi H., Saito K.-S., Tanaka R.
Open Mathematics
|
2013
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tom 11
|
nr 7
1212-1227
EN
Recently, Jiménez-Melado et al. [Jiménez-Melado A., Llorens-Fuster E., Mazcuñán-Navarro E.M., The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 2008, 342(1), 298–310] defined the Dunkl-Williams constant DW(X) of a normed linear space X. In this paper we present some characterizations of this constant. As an application, we calculate DW(ℓ2-ℓ∞) in the Day-James space ℓ2-ℓ∞.
4
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Tight Embeddability of Proper and Stable Metric Spaces

81%
Baudier F., Lancien G.
Analysis and Geometry in Metric Spaces
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2015
|
tom 3
|
nr 1
EN
We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for p ∈ [1,∞], every proper subset of Lp is almost Lipschitzly embeddable into a Banach space X if and only if X contains uniformly the ℓpn’s. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.
5
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Another approach to characterizations of generalized triangle inequalities in normed spaces

81%
Izumida T., Mitani K.-I., Saito K.-S.
Open Mathematics
|
2014
|
tom 12
|
nr 11
1615-1623
EN
In this paper, we consider a generalized triangle inequality of the following type: $$\left\| {x_1 + \cdots + x_n } \right\|^p \leqslant \frac{{\left\| {x_1 } \right\|^p }} {{\mu _1 }} + \cdots + \frac{{\left\| {x_2 } \right\|^p }} {{\mu _n }}\left( {for all x_1 , \ldots ,x_n \in X} \right),$$ where (X, ‖·‖) is a normed space, (µ1, ..., µn) ∈ ℝn and p > 0. By using ψ-direct sums of Banach spaces, we present another approach to characterizations of the above inequality which is given by [Dadipour F., Moslehian M.S., Rassias J.M., Takahasi S.-E., Nonlinear Anal., 2012, 75(2), 735–741].
6
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Characterization of intermediate values of the triangle inequality II

62%
Sano H., Izumida T., Mitani K.-I., Ohwada T., Saito K.-S.
Open Mathematics
|
2014
|
tom 12
|
nr 5
778-786
EN
In [Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035] there was established a norm inequality which characterizes all intermediate values of the triangle inequality, i.e. C n that satisfy 0 ≤ C n ≤ Σj=1n ‖x j‖ − ‖Σj=1n x j‖, x 1,...,x n ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.
7
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Daugavet centers and direct sums of Banach spaces

62%
Bosenko T.
Open Mathematics
|
2010
|
tom 8
|
nr 2
346-356
EN
A linear continuous nonzero operator G: X → Y is a Daugavet center if every rank-1 operator T: X → Y satisfies ||G + T|| = ||G|| + ||T||. We study the case when either X or Y is a sum X 1⊕F X 2 of two Banach spaces X 1 and X 2 by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces X 1 and X 2 there exists a Daugavet center acting from X 1⊕F X 2, and the class of those F such that for some pair of spaces X 1 and X 2 there is a Daugavet center acting into X 1⊕F X 2. We also present several examples of such Daugavet centers.
8
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G-narrow operators and G-rich subspaces

62%
Ivashyna T.
Open Mathematics
|
2013
|
tom 11
|
nr 9
1677-1688
EN
Let X and Y be Banach spaces. An operator G: X → Y is a Daugavet center if ‖G +T‖ = ‖G‖+‖T‖ for every rank-1 operator T. For every Daugavet center G we consider a certain set of operators acting from X, so-called G-narrow operators. We prove that if J is the natural embedding of Y into a Banach space E, then E can be equivalently renormed so that an operator T is (J ○ G)-narrow if and only if T is G-narrow. We study G-rich subspaces of X: Z ⊂ X is called G-rich if the quotient map q: X → X/Z is G-narrow.
9
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Separability of Real Normed Spaces and Its Basic Properties

62%
Nakasho K., Endou N.
Formalized Mathematics
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2015
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tom 23
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nr 1
59-65
EN
In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the real numbers. In the second section, the basic properties of the separable normed spaces are discussed. It is applied to isomorphic spaces via bounded linear operators and double dual spaces. In the last section, it is proved that the completeness and reflexivity are transferred to sublinear normed spaces. The formalization is based on [34], and also referred to [7], [14] and [16].
10
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Cyclic pairs and common best proximity points in uniformly convex Banach spaces

62%
Gabeleh M., Julia Mary P., Eldred Eldred A. A., Olela Otafudu O.
Open Mathematics
|
2017
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tom 15
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nr 1
711-723
EN
In this article, we survey the existence, uniqueness and convergence of a common best proximity point for a cyclic pair of mappings, which is equivalent to study of a solution for a nonlinear programming problem in the setting of uniformly convex Banach spaces. Finally, we provide an extension of Edelstein’s fixed point theorem in strictly convex Banach spaces. Examples are given to illustrate our main conclusions.
11
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Sums of SCD sets and their applications to SCD operators and narrow operators

62%
Kadets V., Shepelska V.
Open Mathematics
|
2010
|
tom 8
|
nr 1
129-134
EN
We answer two open questions concerning the recently introduced notions of slicely countably determined (SCD) sets and SCD operators in Banach spaces. An application to narrow operators in spaces with the Daugavet property is given.
12
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Characterizations of Rotundity and Smoothness by Approximate Orthogonalities

62%
Stypuła T., Wójcik P.
Annales Mathematicae Silesianae
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2016
|
tom 30
|
nr 1
193-201
EN
In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.
13
Content available remote

On local convexity of nonlinear mappings between Banach spaces

52%
Banakh I., Banakh T., Plichko A., Prykarpatsky A.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2264-2271
EN
We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.
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