A NEW PROOF OF $\ell_1\stackrel{\vee}{\bigotimes} X\subseteq c_0\stackrel{\wedge}{\bigotimes}X$

Abstract

In this paper, we first show that for any Banach space $X$, $\ell_1[X]\subseteq\ell_\infty\langle X\rangle$ by Khinchin's inequality. Then by the relationship between Banach-valued sequence spaces and tensor products, we show that for any Banach space $X$, $\ell_1\stackrel{\vee}{\otimes}X\subseteq c_0\stackrel{\wedge}{\otimes}X$.