Characterizations Of Some Kinds Of Proximinal Subspaces Under Generalized Approximations

The study of the best approximation problem in the context of the norm originated in the late 30s of the last century.With the development of modern mathematical branches such as nonlinear analysis and Banach space geometry,research on the best approximation problem based on norm has been paid more and more attention from the end of 1950s.Since the Minkowski functional pc generated by a closed bounded convex subset C of a normed spaces with the origin as an interior point is an extension of norm.Accordingly,the study of the best approximation problem in the context of pc is a natural thing.Since pc is generally not symmetric,this study is meaningful.This essay focuses on two aspects:first,we attempt to generalize some results in the approximation theory in the norm context to the setting of Minkowski functional pC;secondly,we will study the geometric characteristic of the best simultaneous approximation in the condition complete lattice Banach spaces.The research contents and innovations are summarized as follows:In the second chapter,by using the distance formula from a point to a hyperplane under the generalized approximation,the properties(ε*)of subspaces of conjugate space,and the quotient mapping,this paper gives the characterizations ofτC-proximinal subspaces in X under generalized approximation;at the same time,we also illustrate by an example that the results obtained in this paper are an essential extension of corresponding results in the case of norm.In the third chapter,this paper uses the tools set up in the second chapter and the properties similar to the properties(ε*)(i.e.,(k-ε*)and(k-u*)),to give respectively the characterizations for a closed subspace in X to be τC-k-semi-Chebyshev andτC-k-Chebyshv under generalized approximation,which seem to be new in the context of norm.In the meanwhile,we also use generalized approximations to characterize some convexity of the set C.In the fourth chapter,similar to the first two chapters,the thesis will give the characterizations for a closed subspaces of X to be τC-quasi-Chebyshev under generalized approximation by using tools established and the properties(ω-ε*)of subspaces in the conjugate space.In the fifth chapter,this paper points out that the geometric characterizations of the best simultaneous approximation to bounded sets from closed convex sets in a conditional complete lattice Banach space in[14]is wrong,and gives the correct version.We also give a new proof of a characterization of the best simultaneous approximation to a bounded set from a closed convex set in such a space.