Some Kinds Of K-Convexities In Banach Space

This paper consists of three parts.In the first chapter, we introduce the property locally k drop convexity ,give the equivalent condition of the property k-drop convexity and locally k drop convexity ,obtain the slice of unit of them and the quotient space of k-drop convexity.They are the good supplement of the result in reference[1] .In the second chapter ,we definite the Banach space of weakly k drop convexity and k-nearly drop convexity, discuss their properties.In the third chapter,we obtain the convex closure representation of locally k-convexity,weakly locally k-convexity,compact locally k-convexity, weakly compact locally k-convexity and locally k-smoothness,weakly locally k-smoothness,compact locally k-smoothness, weakly compact locally k-smoothness, generalize the relative results in [12], get the property approximate compactness of locally k-convexity,weakly locally k-convexity,compact locally k-convexity, weakly compact locally k-convexity .