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 p...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.展开更多
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 topolog...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.展开更多
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 co...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）.展开更多
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 ...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.展开更多
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...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.展开更多
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 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].
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.
In this paper, we prove when these x ∈ <em>l</em><sub>2</sub> with <img src="Edit_f19f1285-6d80-48bb-b69d-6d6112f13051.bmp" alt="" /> , they have the common <em>...In this paper, we prove when these x ∈ <em>l</em><sub>2</sub> with <img src="Edit_f19f1285-6d80-48bb-b69d-6d6112f13051.bmp" alt="" /> , they have the common <em>δ</em> for strongly ball proximinal. By using this property, we can prove the strong ball proximinality of <em>l</em><span style="white-space:nowrap;"><sub>∞</sub></span>(<em>l</em><sub>2</sub>). Also, we show that equable subspace <em>Y</em> of a Banach space <em>X</em> is actually uniform ball proximinality.展开更多
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...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.展开更多
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 ele...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展开更多
基金supported by the National Natural Science Foundation of China(12271344)the Natural Science Foundation of Shanghai(23ZR1425800)。
文摘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.
基金supported by the National Natural Science Foundation of China(11671252)supported by the National Natural Science Foundation of China(11771278)。
文摘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.
文摘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）.
文摘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.
基金supported by National Natural Science Foundation of China(Grant No.11371296)supported by National Natural Science Foundation of China(Grant No.11201160)+4 种基金supported by National Natural Science Foundation of China(Grant No.11471270)Ph.D Programs Foundation of MEC(Grant No.20130121110032)Natural Science Foundation of Fujian Province(Grant No.2012J05006)Natural Science Foundation of Fujian Province(Grant No.2015J01022)supported by NSF(Grant No.DMS-1200370)
文摘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.
文摘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.
文摘 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].
文摘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.
文摘In this paper, we prove when these x ∈ <em>l</em><sub>2</sub> with <img src="Edit_f19f1285-6d80-48bb-b69d-6d6112f13051.bmp" alt="" /> , they have the common <em>δ</em> for strongly ball proximinal. By using this property, we can prove the strong ball proximinality of <em>l</em><span style="white-space:nowrap;"><sub>∞</sub></span>(<em>l</em><sub>2</sub>). Also, we show that equable subspace <em>Y</em> of a Banach space <em>X</em> is actually uniform ball proximinality.
文摘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.
文摘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
