Convexities And Approximation Properties

This paper consists of three parts.In the first chapter,some concepts and various theorems of Bochner integrable fuaction spaces L_p(μ, X) and approximation theoretics are introduced;In the second chapter, we give the lifting of very converxity and extremal converxity in L_p(μ, X) in first part,Radon-Nikodym property in substitution spaces P_XX_n are given in third part ,thus generalize the interest results in [23] ;at last we give the lifting of drop property in L_p(μ, X) through a more simple method; In the third chapter,we discuss the approximation properties in P_bB_S, and the simultaneous approximation in l_v(X_k) .