scholarly journals Convexities and Existence of the Farthest Point

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Z. H. Zhang ◽  
C. Y. Liu
Keyword(s):  
Banach Space ◽  
Closed Subset ◽  
Sufficient Condition ◽  
Residual Subset ◽  
Farthest Points ◽  
Weakly Closed ◽  
Farthest Point ◽  
Farthest Point Map

Five counterexamples are given, which show relations among the new convexities and some important convexities in Banach space. Under the assumption that Banach space is nearly very convex, we give a sufficient condition that bounded, weakly closed subset of has the farthest points. We also give a sufficient condition that the farthest point map is single valued in a residual subset of when is very convex.

scholarly journals On the farthest points in convex metric spaces and linear metric spaces

2014 ◽  
Vol 95 (109) ◽  
pp. 229-238 ◽  
Author(s):  
S Sangeeta ◽  
T.D. Narang
Keyword(s):  
Metric Spaces ◽  
Farthest Points ◽  
Strictly Convex ◽  
Convex Metric Spaces ◽  
Farthest Point ◽  
Farthest Point Map

We prove some results on the farthest points in convex metric spaces and in linear metric spaces. The continuity of the farthest point map and characterization of strictly convex linear metric spaces in terms of farthest points are also discussed.

scholarly journals Monotonicity and the Dominated Farthest Points Problem in Banach Lattice

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
H. R. Khademzadeh ◽  
H. Mazaheri
Keyword(s):  
Orlicz Function ◽  
Orlicz Spaces ◽  
Farthest Points ◽  
Strong Solvability ◽  
Fréchet Differentiability ◽  
Measure Spaces ◽  
Nonatomic Measure ◽  
Frechet Differentiability ◽  
Farthest Point ◽  
Farthest Point Map

We introduce the dominated farthest points problem in Banach lattices. We prove that for two equivalent norms such thatXbecomes an STM and LLUM space the dominated farthest points problem has the same solution. We give some conditions such that under these conditions the Fréchet differentiability of the farthest point map is equivalent to the continuity of metric antiprojection in the dominated farthest points problem. Also we prove that these conditions are equivalent to strong solvability of the dominated farthest points problem. We prove these results in STM, reflexive STM, and UM spaces. Moreover, we give some applications of the stated results in Musielak-Orlicz spacesL ​ϕ(μ)andE ​ϕ(μ)over nonatomic measure spaces in terms of the functionϕ. We will prove that the Fréchet differentiability of the farthest point map and the conditionsϕ∈Δ2andϕ>0in reflexive Musielak-Orlicz function spaces are equivalent.

scholarly journals A farthest-point characterisation of the relative Chebyshev centre

1996 ◽  
Vol 54 (1) ◽  
pp. 27-33 ◽  
Author(s):  
R. Huotari ◽  
M.P. Prophet ◽  
J. Shi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Convex Hull ◽  
Compact Subset ◽  
Real Banach Space ◽  
Metric Projection ◽  
Closed Convex Hull ◽  
Farthest Points ◽  
Farthest Point ◽  
Gateaux Derivative

We characterise the relative Chebyshev centre of a compact subsetFof a real Banach space in terms of the Gateaux derivative of the distance to farthest points. We present a relative-Chebyshev-centre characterisation of Hilbert space. In Hilbert space we show that the relative Chebyshev centre is in the closed convex hull of the metric projection ofF, and we estimate the relative Chebyshev radius ofF.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

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 Weighted Polynomial Approximation With Gaps

2005 ◽  
Vol 178 ◽  
pp. 55-61 ◽  
Author(s):  
Guantie Deng
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Polynomial Approximation ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Weighted Polynomial Approximation ◽  
Necessary And Sufficient Condition ◽  
Weighted Banach Space ◽  
Nonnegative Continuous Function ◽  
Necessary And Sufficient

Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα, where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.

Completions of Boolean algebras of projections and weak-star closures of C*-algebras on dual Banach spaces

2011 ◽  
Vol 54 (2) ◽  
pp. 515-529
Author(s):  
Philip G. Spain
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Boolean Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Weak Star ◽  
Weakly Closed ◽  
Operator Closure ◽  
Dual Banach Space ◽  
Completion Theorem

AbstractPalmer has shown that those hermitians in the weak-star operator closure of a commutative C*-algebra represented on a dual Banach space X that are known to commute with the initial C*-algebra form the real part of a weakly closed C*-algebra on X. Relying on a result of Murphy, it is shown in this paper that this last proviso may be dropped, and that the weak-star closure is even a W*-algebra.When the dual Banach space X is separable, one can prove a similar result for C*-equivalent algebras, via a ‘separable patch’ completion theorem for Boolean algebras of projections on such spaces.

scholarly journals Existence of approximate solution to abstract nonlocal Cauchy problem

1992 ◽  
Vol 5 (4) ◽  
pp. 363-373 ◽  
Author(s):  
L. Byszewski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Approximate Solution ◽  
Closed Subset ◽  
Nonlocal Condition ◽  
Right Hand ◽  
The Right ◽  
Nonlocal Cauchy Problem

The aim of the paper is to prove a theorem about the existence of an approximate solution to an abstract nonlinear nonlocal Cauchy problem in a Banach space. The right-hand side of the nonlocal condition belongs to a locally closed subset of a Banach space. The paper is a continuation of papers [1], [2] and generalizes some results from [3].

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