Continuity And Convergence Properties Of Metric Projection

In this paper, we study the relations between the propeities (C - II) and approximate compactness and define new properties (N - K) and (N - WM), which are the limit forms of properties (C - K) and (C - WM). While we discuss the relations between the properties (N-K) and approximate compactness,we get some equivalent conditions about the property (N-II) .We prove that LwR(CLwR; wCLwR) has the property (N-I)((N-II); (N-III)).Meanwhile,we improve the results of parper [14].Deleting the condition of rcflexivity,we get a convergence result of metric projection in a nonreflexive Banach space.We prove that if Banach space has property (F), {Cn} is non-empty closed convex subsets of B, Cn is Chebyshev set n = 1,2, ... , if x B, {Pcn(x)} is norm converging, s - limCn isChebyshev set,then there existing C which is non-empty closed convex subsets of B such that Cn - M C,PCn{x) - PC{x),x B.Besides those, we improve the results of paper [23].Deleting the condition of rcflcxivity,we get a character of metric projection in general Banach spaces ,making it use more widely.Last ,we discuss approximate compactness and weakly approximate compactness in PBBs spaces , our work generalises some results of the corresponding papers.