Banach Space Geometry And The Best Approximation

Approximation theory has been considered as one of the most important branch of modern mathematics.It plays a very important part in the study of mathematical theories and practical applications.In the late eighteenth,Jefimocv and Stechkin introduced a conception which is approximate compactness and Russian mathematician Chebyshev proposed the conception of optimal approximation.Up to now,the research of the nonlinear approximate theories is still on studying.The theory of optimal approximation is one of the most important part of the best approximation theory and a research over Banach spaces in usual.The questions about the existence and uniqueness of the best approximation element always depend on the structure of the space,such as its convexity,smoothness and function-differentiability.In addition,the continuity of metric projection is a significant factor too.Therefore,the geometric property is vital in approximation theory studying.In this section,we introduce two new property in Banach space which are property Q and property WQ,and discuss the relation between this property and approximation from geometric property and structure.In the first chapter,we introduce the research background of this thesis and relative basic concepts and results.In the second chapter,we discuss the connection between the property Q(WQ)and the geometric structure.And we prove that the dusmooth space(vary smooth space)if and only if X has property Q(WQ).We also prove al space X of X is a strongly that any closed convex set is(weakly)approximately compact Chebyshev set if and only if X has property Q(WQ).Finally,we discuss the connection with property H,property S,reflexivity and near concavity.In the third chapter,we discuss the connection between property Q(WQ)and property(C-k)and Locally asymptotic-norming property.We prove that Banach space X with property Q(WQ)is equivalent with Banach space X with B(X*)-LANP-?(?')over reflexivity.Moreover,we also prove that the continuity of the metric projection is continuous(weak continuous)if Banach space has property Q(WQ).