In this paper, first, we consider closed convex and bounded subsets of infinite-dimensional unital Banach algebras and show with regard to the general conditions that these sets are not quasi-Chebyshev and pseudo-Cheb...In this paper, first, we consider closed convex and bounded subsets of infinite-dimensional unital Banach algebras and show with regard to the general conditions that these sets are not quasi-Chebyshev and pseudo-Chebyshev. Examples of those algebras are given including the algebras of continuous functions on compact sets. We also see some results in C＊-algebras and Hilbert C＊-modules. Next, by considering some conditions, we study Chebyshev of subalgebras in C＊-algebras.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
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 this article, we prove the following results： （1） A Banach space X is weak midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively weakly compact k-Chebyshev set; （2） A ...In this article, we prove the following results： （1） A Banach space X is weak midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively weakly compact k-Chebyshev set; （2） A Banach space X is midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively compact k-Chebyshev set.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
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
文摘In this paper, first, we consider closed convex and bounded subsets of infinite-dimensional unital Banach algebras and show with regard to the general conditions that these sets are not quasi-Chebyshev and pseudo-Chebyshev. Examples of those algebras are given including the algebras of continuous functions on compact sets. We also see some results in C＊-algebras and Hilbert C＊-modules. Next, by considering some conditions, we study Chebyshev of subalgebras in C＊-algebras.
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
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supported by the National Natural Science Foundation of China(11671252)supported by the National Natural Science Foundation of China(11771278)
文摘In this article, we prove the following results： （1） A Banach space X is weak midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively weakly compact k-Chebyshev set; （2） A Banach space X is midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively compact k-Chebyshev set.
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.
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.
