scholarly journals On geometric properties of ⊥BJCϵ symmetric in some types of Banach spaces

2020 ◽  
Vol 1664 (1) ◽  
pp. 012038
Author(s):  
Saied A. Jhonny ◽  
Buthainah A. A. Ahmed
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Real Banach Space ◽  
Convex Banach Space ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Geometric Properties ◽  
Strictly Convex ◽  
Strictly Convex Banach Space

Abstract In this paper, we ⊥ B J C ϵ -orthogonality and explore ⊥ B J C ϵ -symmetricity such as a ⊥ B J C ϵ -left-symmetric ( ⊥ B J C ϵ -right-symmetric) of a vector x in a real Banach space (𝕏, ‖·‖𝕩) and study the relation between a ⊥ B J C ϵ -right-symmetric ( ⊥ B J C ϵ -left-symmetric) in ℐ(x). New results and proofs are include the notion of norm attainment set of a continuous linear functionals on a reflexive and strictly convex Banach space and using these results to characterize a smoothness of a vector in a unit sphere.

NONEXPANSIVE BIJECTIONS TO THE UNIT BALL OF THE -SUM OF STRICTLY CONVEX BANACH SPACES

2018 ◽  
Vol 97 (2) ◽  
pp. 285-292 ◽  
Author(s):  
V. KADETS ◽  
O. ZAVARZINA
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Convex Banach Space ◽  
Strictly Convex ◽  
Strictly Convex Banach Space ◽  
Strictly Convex Banach Spaces

Extending recent results by Cascales et al. [‘Plasticity of the unit ball of a strictly convex Banach space’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.110(2) (2016), 723–727], we demonstrate that for every Banach space $X$ and every collection $Z_{i},i\in I$, of strictly convex Banach spaces, every nonexpansive bijection from the unit ball of $X$ to the unit ball of the sum of $Z_{i}$ by $\ell _{1}$ is an isometry.

Common best proximity pairs in strictly convex Banach spaces

2017 ◽  
Vol 24 (3) ◽  
pp. 363-372 ◽  
Author(s):  
Moosa Gabeleh
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Banach Space ◽  
Existence Results ◽  
Current Paper ◽  
Strictly Convex ◽  
Space Setting ◽  
Strictly Convex Banach Space ◽  
Strictly Convex Banach Spaces ◽  
Best Proximity Pair

AbstractA mapping {T\colon A\cup B\to A\cup B} such that {T(A)\subseteq A} and {T(B)\subseteq B} is called a noncyclic mapping, where A and B are two nonempty subsets of a Banach space X. A best proximity pair {(p,q)\in A\times B} for such a mapping T is a point such that {p=Tp,q=Tq} and {\|p-q\|=\operatorname{dist}(A,B)}. In the current paper, we establish some existence results of best proximity pairs in strictly convex Banach spaces. The presented theorems improve and extend some recent results in the literature. We also obtain a generalized version of Markov–Kakutani’s theorem for best proximity pairs in a strictly convex Banach space setting.

scholarly journals Range-Kernel orthogonality and elementary operators on certain Banach spaces

2021 ◽  
Vol 54 (1) ◽  
pp. 272-279
Author(s):  
Ahmed Bachir ◽  
Abdelkader Segres ◽  
Nawal Ali Sayyaf ◽  
Khalid Ouarghi
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Banach Space ◽  
Strictly Convex ◽  
Strictly Convex Banach Space ◽  
Arbitrary Operator ◽  
Elementary Operators

Abstract The characterization of the points in C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , the Von Neuman-Schatten p-classes, that are orthogonal to the range of elementary operators has been done for certain kinds of elementary operators. In this paper, we shall study this problem of characterization on an abstract reflexive, smooth and strictly convex Banach space for arbitrary operator. As an application, we consider other kinds of elementary operators defined on the spaces C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , and finally, we give a counterexample to Mecheri’s result given in this context.

Projections in the Convex Hull of Surjective Isometries

2010 ◽  
Vol 53 (3) ◽  
pp. 398-403 ◽  
Author(s):  
Fernanda Botelho ◽  
James Jamison
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Spaces ◽  
Convex Combination ◽  
Continuous Functions ◽  
Convex Banach Space ◽  
Strictly Convex ◽  
Strictly Convex Banach Space ◽  
Spaces Of Continuous Functions

AbstractWe characterize those linear projections represented as a convex combination of two surjective isometries on standard Banach spaces of continuous functions with values in a strictly convex Banach space.

scholarly journals Properties of Chmielinski-orthogonality using Kadets-Klee property

2022 ◽  
pp. 94-101
Author(s):  
saied Johnny ◽  
Buthainah A. A. Ahmed
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Real Banach Space ◽  
Convex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Strictly Convex ◽  
Strictly Convex Banach Space ◽  
James Orthogonality ◽  
New Type

The aim of this paper is to study new results of an approximate orthogonality of Birkhoff-James techniques in real Banach space , namely Chiemelinski orthogonality (even there is no ambiguity between the concepts symbolized by orthogonality) and provide some new geometric characterizations which is considered as the basis of our main definitions. Also, we explore relation between two different types of orthogonalities. First of them orthogonality in a real Banach space and the other orthogonality in the space of bounded linear operator . We obtain a complete characterizations of these two orthogonalities in some types of Banach spaces such as strictly convex space, smooth space and reflexive space. The study is designed to give different results about the concept symmetry of Chmielinski-orthogonality for a compact linear operator on a reflexive, strictly convex Banach space having Kadets-Klee property by exploring a new type of a generalized some results with Birkhoff James orthogonality in the space of bounded linear operators. We also exhibit a smooth compact linear operator with a spectral value that is defined on a reflexive, strictly convex Banach space having Kadets-Klee property either having zero nullity or not -right-symmetric.

A Non-Reflexive Banach Space has Non-Smooth Third Conjugate Space

1974 ◽  
Vol 17 (1) ◽  
pp. 117-119 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Unit Sphere ◽  
Reflexive Banach Space ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Natural Embedding ◽  
Conjugate Space ◽  
The Third

J. Dixmier has observed [3, p. 1070] that a non-reflexive Banach space has non-rotund fourth conjugate space. It is the aim of this paper to improve Dixmier’s result by showing that a non-reflexive Banach space already has non-smooth third conjugate space in that the images under natural embedding of the continuous linear functionals which do not attain their norm on the unit sphere are non-smooth points of the third conjugate space.

scholarly journals The Denjoy–Wolff Theorem in the Open Unit Ball of a Strictly Convex Banach Space

1999 ◽  
Vol 143 (1) ◽  
pp. 111-123 ◽  
Author(s):  
Jaroslaw Kapeluszny ◽  
Tadeusz Kuczumow ◽  
Simeon Reich
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Convex Banach Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Strictly Convex ◽  
Strictly Convex Banach Space

Conjugate Banach spaces

1957 ◽  
Vol 53 (3) ◽  
pp. 576-580 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Banach Space ◽  
Linear Mapping ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Linear Mapping ◽  
Conjugate Space ◽  
Necessary And Sufficient ◽  
Weak Topologies

The purpose of this note is to present two characterizations of conjugate Banach spaces. More precisely, we present two conditions, each necessary and sufficient for a (real or complex) Banach space to be isomorphic to the conjugate space of a Banach space, and two corresponding conditions for to be equivalent to the conjugate space of a Banach space. Other characterizations, in terms of weak topologies, have been given by Alaoglu ((1), Theorem 2:1, p. 256, and Corollary 2:1, p. 257) and Bourbaki ((4), Chap, IV, §5, exerc. 15c, p. 122). Here, by the conjugate space* of a Banach space we mean ((2), p. 188) the space of continuous linear functionals over . Two Banach spaces and are said to be isomorphic if there is a one-one continuous linear mapping of onto (its inverse is necessarily continuous by the inversion theorem ((2), Théorème 5, p. 41; (6), Theorem 2·13·7, Corollary, p. 29)); they are said to be equivalent if there is a norm-preserving linear mapping of onto .((2), p. 180).

scholarly journals Strong convergence of a modified Halpern-type iteration for asymptotically quasi-∅-nonexpansive mappings

2013 ◽  
Vol 21 (1) ◽  
pp. 261-276
Author(s):  
Chang-Qun Wu ◽  
Yan Hao
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Convex Banach Space ◽  
Halpern Iteration ◽  
Uniformly Smooth ◽  
Strictly Convex Banach Space

Abstract In this paper, the problem of modifying Halpern iteration for approximating a common fixed point of a family of asymptotically quasi- ∅-nonexpansive mappings is considered. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper mainly improve the corresponding results announced in [Y.J. Cho, X. Qin, S.M. Kang, Strong convergence of the modified Halpern- type iterative algorithms in Banach spaces, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 17 (2009) 51-68].

scholarly journals Isometries between Spaces of Vector-Valued Differentiable Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Alireza Ranjbar-Motlagh
Keyword(s):  
Banach Space ◽  
Real Line ◽  
Convex Banach Space ◽  
Compact Interval ◽  
The Real ◽  
Strictly Convex ◽  
Differentiable Functions ◽  
Strictly Convex Banach Space ◽  
Vector Valued

This article characterizes the isometries between spaces of all differentiable functions from a compact interval of the real line into a strictly convex Banach space.

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