SOME EQUIVALENT CONDITIONS OF PROXIMINALITY IN NONREFLEXIVE BANACH SPACES

In this paper,we discuss the relation between τ-strongly Chebyshev,approximatively τ-compact k-Chebyshev,approximatively τ-compact,τ-strongly proximinal and proximinal sets,where τ is the norm or the weak topology.We give some equivalent conditions regarding the above proximinality.Furthermore,we also propose the necessary and sufficient conditions that a half-space is τ-strongly proximinal,approximatively τ-compact and τ-strongly Chebyshev.

CHEBYSHEV CENTERS PROXIMINALITY AND FARTHEST POINTS IN STRONG NORMED ALMOST LINEAR SPACES

Some results from the theory of best (or best simultaneous) approximation in a narmed linear space have been extended to a normed almost linear space [strong normed almost linear space].

A Note on Ball Proximinality

In this paper, we prove when these x ∈ l2 with , they have the common δ for strongly ball proximinal. By using this property, we can prove the strong ball proximinality of l∞(l2). Also, we show that equable subspace Y of a Banach space X is actually uniform ball proximinality.

THREE KINDS OF DENTABILITIES IN BANACH SPACES AND THEIR APPLICATIONS

In this paper,we study some dentabilities in Banach spaces which are closely related to the famous Radon-Nikodym property.We introduce the concepts of the weak^(*)-weak denting point and the weak^(*)-weak^(*)denting point of a set.These are the generalizations of the weak^(*)denting point of a set in a dual Banach space.By use of the weak^(*)-weak denting point,we characterize the very smooth space,the point of weak^(*)-weak continuity,and the extreme point of a unit ball in a dual Banach space.Meanwhile,we also characterize an approximatively weak compact Chebyshev set in dual Banach spaces.Moreover,we define the nearly weak dentability in Banach spaces,which is a generalization of near dentability.We prove the necessary and sufficient conditions of the reflexivity by nearly weak dentability.We also obtain that nearly weak dentability is equivalent to both the approximatively weak compactness of Banach spaces and the w-strong proximinality of every closed convex subset of Banach spaces.

On Proximinality of Convex Sets in Superspaces

Abstract In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal （proximinal, resp.） in every Banach space isometrically containing it if and only if C is locally （weakly, resp.） compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.

ON SUM AND QUOTIENT OF QUASI-CHEBYSHEV SUBSPACES IN BANACH SPACES

It will be determined under what conditions types of proximinality are transmitted to and from quotient spaces. In the final section, by many examples we show that types of proximinality of subspaces in Banach spaces can not be preserved by equivalent norms.

SOME APPLICATIONS OF BP-THEOREM IN APPROXIMATION THEORY

In this paper we apply Bishop-Phelps property to show that if X is a Banach space and G _ X is the maximal subspace so that G⊥ ： {x＊ ∈ X＊ ｜x＊ （y） = 0; y ∈ G} is an L-summand in X＊, then L1 （Ω, G） is contained in a maximal proximinal subspace of L1（Ω,X）.

SOME RESULTS ON NEAREST POINTS IN NORMED LINEAR SPACES

We will focus on some results that we hope to give an algorithm for constructing the best approximations in some types of normed linear spaces. Also some results on best approximation will be obtained.

Some Results on the Simultaneous Approximation

In this paper, we give some result on the simultaneous proximinal subset and simultaneous Chebyshev in the uniformly convex Banach space. Also we give relation between fixed point theory and simultaneous proximity.

PROXIMINAL SUBSPACES OF 2-NORMED SPACES

In 1965 Gohler introduced 2-normed spaces and since then, this topic have been intensively studied and developed. We shall introduce the notion of 1-type proximinal subspaces of 2-normed spaces and give some results in this field.

赋范空间中闭凸集族的正则性条件

考虑在赋范空间中,具有非空交集的闭凸集族的正则性条件,并利用集值映射来证明在Banach空间中存在Proximinal集的集族的核非空且集族交集有界时,正则性也存在.

BEST APPROXIMATION BY DOWNWARD SETS WITH APPLICATIONS

We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs （W,x）, where ∈ X and W is a closed downward subset of X

CHARACTERIZATION OF BEST APPROXIMATIONS IN METRIC LINEAR SPACES

Let (X,d) be a real metric linear space, with translation-invariant metric d and C a linear subspace of X. In this paper we use functionals in the Lipschitz dual of X to characterize those elements of G which are best approximations to elements of X.We also give simultaneous characterization of elements of best approximation and also consider elements of ε-approximation.