scholarly journals On Simultaneous Farthest Points in

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Sh. Al-Sharif ◽  
M. Rawashdeh
Keyword(s):  
Banach Space ◽  
Bounded Subset ◽  
Farthest Points ◽  
Integrable Functions

Let be a Banach space and let be a closed bounded subset of . For , we set  . The set is called simultaneously remotal if, for any , there exists such that  . In this paper, we show that if is separable simultaneously remotal in , then the set of -Bochner integrable functions, , is simultaneously remotal in . Some other results are presented.

scholarly journals On farthest and simultaneous farthest points in Kothe spaces

2012 ◽  
Vol 21 (2) ◽  
pp. 123-128
Author(s):  
SH. AL-SHARIF ◽  
Keyword(s):  
Banach Space ◽  
Function Space ◽  
Real Banach Space ◽  
Bounded Subset ◽  
Farthest Points ◽  
Köthe Spaces

Let X be a real Banach space and G be a closed bounded subset of X. For x e X, we set ρ (x, G) = sup {kx − yk : y e G} . The set G is called remotal if for any x e X, there exists g e G such that ρ (x, G) = kx − gk . In this paper we show that for a separable remotal set G ⊂ X, the set E(G) is remotal in E(X), where E(X) is the Kothe Bochner function space. Some other results are presented.

Perturbation of the fixed point problem for continuous mappings

2020 ◽  
pp. 241-249
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Continuous Mapping ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Bounded Subset ◽  
Completely Continuous ◽  
Continuous Map ◽  
Continuous Mappings

We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.

scholarly journals Best Simultaneous Approximation in Orlicz Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
M. Khandaqji ◽  
Sh. Al-Sharif
Keyword(s):  
Banach Space ◽  
Simultaneous Approximation ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Unit Interval ◽  
Luxemburg Norm ◽  
Integrable Functions ◽  
Best Simultaneous Approximation ◽  
Distance Formula

LetXbe a Banach space and letLΦ(I,X)denote the space of OrliczX-valued integrable functions on the unit intervalIequipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, whereGis a closed subspace ofX, andf1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation inLΦ(I,X)are presented.

On Asymptotic Centers and Fixed Points of Nonexpansive Mappings

1980 ◽  
Vol 32 (2) ◽  
pp. 421-430 ◽  
Author(s):  
Teck-Cheong Lim
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Nonempty Subset ◽  
Nonexpansive Mappings ◽  
Chebyshev Center ◽  
Bounded Subset ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Asymptotic Centers ◽  
Image Position

Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define

Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions

1977 ◽  
Vol 29 (5) ◽  
pp. 963-970 ◽  
Author(s):  
Mark A. Smith
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Nonzero Element ◽  
Bounded Subset ◽  
Compact Sets ◽  
Weakly Compact ◽  
Uniform Rotundity ◽  
Minimum Depth ◽  
Weakly Compact Sets

In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.

scholarly journals Injective absolutely summing operators

1988 ◽  
Vol 103 (3) ◽  
pp. 497-502
Author(s):  
Susumu Okada ◽  
Yoshiaki Okazaki
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Convex Space ◽  
Locally Convex Space ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Integrable Functions ◽  
Locally Convex ◽  
Convergence In Mean

Let X be an infinite-dimensional Banach space. It is well-known that the space of X-valued, Pettis integrable functions is not always complete with respect to the topology of convergence in mean, that is, the uniform convergence of indefinite integrals (see [14]). The Archimedes integral introduced in [9] does not suffer from this defect. For the Archimedes integral, functions to be integrated are allowed to take values in a locally convex space Y larger than the space X while X accommodates the values of indefinite integrals. Moreover, there exists a locally convex space Y, into which X is continuously embedded, such that the space ℒ(μX, Y) of Y-valued, Archimedes integrable functions is identical to the completion of the space of X valued, simple functions with repect to the toplogy of convergence in mean, for each non-negative measure μ (see [9]).

scholarly journals Non-weak compactness of the integration map for vector measures

1993 ◽  
Vol 54 (3) ◽  
pp. 287-303 ◽  
Author(s):  
S. Okada ◽  
W. J. Ricker
Keyword(s):  
Banach Space ◽  
Volterra Operator ◽  
Continuous Operator ◽  
Weakly Compact ◽  
Vector Measures ◽  
Mean Convergence ◽  
Integrable Functions ◽  
The Mean ◽  
The Fourier Transform ◽  
Adjoint Operators

AbstractLet m be a vector measure with values in a Banach space X. If L1(m) denotes the space of all m integrable functions then, with respect to the mean convergence topology, L1(m) is a Banach space. A natural operator associated with m is its integration map Im which sends each f of L1(m) to the element ∫fdm (of X). Many properties of the (continuous) operator Im are closely related to the nature of the space L1(m). In general, it is difficult to identify L1(m). We aim to exhibit non-trivial examples of measures m in (non-reflexive) spaces X for which L1(m) can be explicitly computed and such that Im is not weakly compact. The examples include some well known operators from analysis (the Fourier transform on L1 ([−π, π]), the Volterra operator on L1 ([0, 1]), compact self-adjoint operators in a Hilbert space); such operators can be identified with integration maps Im (or their restrictions) for suitable measures m.

scholarly journals Farthest points and the farthest distance map

2005 ◽  
Vol 71 (3) ◽  
pp. 425-433 ◽  
Author(s):  
Pradipta Bandyopadhyay ◽  
S. Dutta
Keyword(s):  
Banach Space ◽  
Maximal Monotone ◽  
Convex Banach Space ◽  
Closed Convex Hull ◽  
Compact Sets ◽  
Weakly Compact ◽  
Distance Map ◽  
Farthest Points ◽  
Bounded Set ◽  
Weakly Compact Sets

In this paper, we consider farthest points and the farthest distance map of a closed bounded set in a Banach space. We show, inter alia, that a strictly convex Banach space has the Mazur intersection property for weakly compact sets if and only if every such set is the closed convex hull of its farthest points, and recapture a classical result of Lau in a broader set-up. We obtain an expression for the subdifferential of the farthest distance map in the spirit of Preiss' Theorem which in turn extends a result of Westphal and Schwartz, showing that the subdifferential of the farthest distance map is the unique maximal monotone extension of a densely defined monotone operator involving the duality map and the farthest point map.

scholarly journals Best simultaneous approximation on metric spaces via monotonous norms

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3777-3787
Author(s):  
Mona Khandaqji ◽  
Aliaa Burqan
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Finite Number ◽  
Metric Space ◽  
Measure Space ◽  
Simultaneous Approximation ◽  
Metric Spaces ◽  
Closed Subspace ◽  
Integrable Functions ◽  
Best Simultaneous Approximation

For a Banach space X, L?(T,X) denotes the metric space of all X-valued ?-integrable functions f : T ? X, where the measure space (T,?,?) is a complete positive ?-finite and ? is an increasing subadditive continuous function on [0,?) with ?(0) = 0. In this paper we discuss the proximinality problem for the monotonous norm on best simultaneous approximation from the closed subspace Y?X to a finite number of elements in X.

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