The Kadec-Peł czynski theorem in $L^p$, $1\le p<2$

Abstract

By a classical result of Kadec and Pełczynski (1962), every normalized weakly null sequence in $L^p$, $p>2$ contains a subsequence equivalent to the unit vector basis of ${\ell }^2$ or to the unit vector basis of ${\ell }^p$. In this paper we investigate the case $1\le p<2$ and show that a necessary and sufficient condition for the first alternative in the Kadec-Pełczynski theorem is that the limit random measure $\mu$ of the sequence satisfies ${\int }_{\mathbb{R}}{x}^{2}d\mu (x)\in L^{p/2}$.