Based on an application of the Davis-Figiel-Johnson-Pelzyski procedure, this note shows that every weakly compact subset of a Banach space can be uniformly embedded into a reflexive Banach space. As its application, w...Based on an application of the Davis-Figiel-Johnson-Pelzyski procedure, this note shows that every weakly compact subset of a Banach space can be uniformly embedded into a reflexive Banach space. As its application, we present the recent renorming theorems for reflexive spaces of Odell- Schlumprecht and Hajek-Johanis can be extended and localized to weakly compact convex subsets of an arbitrary Banach space.展开更多
The concepts of quasi-Chebyshev and weakly-Chebyshev and σ-Chebyshev were defined [3 - 7], andas a counterpart to best approximation in normed linear spaces, best coapprozimation was introduced by Franchetti and Furi...The concepts of quasi-Chebyshev and weakly-Chebyshev and σ-Chebyshev were defined [3 - 7], andas a counterpart to best approximation in normed linear spaces, best coapprozimation was introduced by Franchetti and Furi^[1]. In this research, we shall define τ-Chebyshev subspaces and τ-cochebyshev subspaces of a Banach space, in which the property τ is compact or weakly-compact, respectively. A set of necessary and sufficient theorems under which a subspace is τ-Chebyshev is defined.展开更多
In the framework of topological vector spaces, we give a characterization of strong Minkowski separation, introduced by Cheng, et al., in terms of convex body separation. From this, several results on strong Minkowski...In the framework of topological vector spaces, we give a characterization of strong Minkowski separation, introduced by Cheng, et al., in terms of convex body separation. From this, several results on strong Minkowski separation are deduced. Using the results, we prove a drop theorem involving weakly countably compact sets in locally convex spaces. Moreover, we introduce the notion of the co-drop property and show that every weakly countably compact set has the co-drop property. If the underlying locally convex space is quasi-complete, then a bounded weakly closed set has the co-drop property if and only if it is weakly countably compact.展开更多
Let G be a nonempty closed subset of a Banach space X.Let B(X)be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and B_(G)(X)={A∈B(X):A∩G=φ},where the closure is taken in the ...Let G be a nonempty closed subset of a Banach space X.Let B(X)be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and B_(G)(X)={A∈B(X):A∩G=φ},where the closure is taken in the metric space(B(X),H).For x∈X and F∈B_(G)(X),we denote the nearest point problem inf{||x-g||:g∈G}by min(x,G)and the mutually nearest point problem inf{||f-g||:f∈ F,g∈G}by min(F,G).In this paper,parallel to well-posedness of the problems min(a:,G)and mm(F,G)which are defined by De Blasi et al.,we further introduce the weak well-posedness of the problems min(x,G)and min(F,G).Under the assumption that the Banach space X has some geometric properties,we prove a series of results on weak well-posedness of min(x,G)and min(F,G).We also give two sufficient conditions such that two classes of subsets of X are almost Chebyshev sets.展开更多
文摘Based on an application of the Davis-Figiel-Johnson-Pelzyski procedure, this note shows that every weakly compact subset of a Banach space can be uniformly embedded into a reflexive Banach space. As its application, we present the recent renorming theorems for reflexive spaces of Odell- Schlumprecht and Hajek-Johanis can be extended and localized to weakly compact convex subsets of an arbitrary Banach space.
文摘The concepts of quasi-Chebyshev and weakly-Chebyshev and σ-Chebyshev were defined [3 - 7], andas a counterpart to best approximation in normed linear spaces, best coapprozimation was introduced by Franchetti and Furi^[1]. In this research, we shall define τ-Chebyshev subspaces and τ-cochebyshev subspaces of a Banach space, in which the property τ is compact or weakly-compact, respectively. A set of necessary and sufficient theorems under which a subspace is τ-Chebyshev is defined.
基金National Natural Science Foundation of China(10571035)
文摘In the framework of topological vector spaces, we give a characterization of strong Minkowski separation, introduced by Cheng, et al., in terms of convex body separation. From this, several results on strong Minkowski separation are deduced. Using the results, we prove a drop theorem involving weakly countably compact sets in locally convex spaces. Moreover, we introduce the notion of the co-drop property and show that every weakly countably compact set has the co-drop property. If the underlying locally convex space is quasi-complete, then a bounded weakly closed set has the co-drop property if and only if it is weakly countably compact.
基金Supported by the NSFC(Grant No.11671252)the NSFC(Grant No.11771278)。
文摘Let G be a nonempty closed subset of a Banach space X.Let B(X)be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and B_(G)(X)={A∈B(X):A∩G=φ},where the closure is taken in the metric space(B(X),H).For x∈X and F∈B_(G)(X),we denote the nearest point problem inf{||x-g||:g∈G}by min(x,G)and the mutually nearest point problem inf{||f-g||:f∈ F,g∈G}by min(F,G).In this paper,parallel to well-posedness of the problems min(a:,G)and mm(F,G)which are defined by De Blasi et al.,we further introduce the weak well-posedness of the problems min(x,G)and min(F,G).Under the assumption that the Banach space X has some geometric properties,we prove a series of results on weak well-posedness of min(x,G)and min(F,G).We also give two sufficient conditions such that two classes of subsets of X are almost Chebyshev sets.
