The weak Banach-Saks Property of the Space $(L_{\mu }^p{)}^m$

Abstract

In this paper we show the weak Banach-Saks property of the Banach vector space $(L_{\mu }^p{)}^m$ generated by $m$ $L_{\mu }^p$-spaces for $1\le p<+\infty ,$ where $m$ is any given natural number. When $m=1,$ this is the famous Banach-Saks-Szlenk theorem. By use of this property, we also present inequalities for integrals of functions that are the composition of nonnegative continuous convex functions on a convex set of a vector space ${\mathbf{R}}^m$ and vector-valued functions in a weakly compact subset of the space $(L_{\mu }^p{)}^m$ for $1\le p<+\infty$ and inequalities when these vector-valued functions are in a weakly* compact subset of the product space $(L_{\mu }^{\infty }{)}^m$ generated by $m$ $L_{\mu }^{\infty }$-spaces.

arXiv comment7