Small subspaces of $L_p$

Abstract

We prove that if $X$ is a subspace of $L_p$ $(2\lt p\lt \infty)$, then either $X$ embeds isomorphically into $\ell_p \oplus \ell_2$ or $X$ contains a subspace $Y,$ which is isomorphic to $\ell_p(\ell_2)$. We also give an intrinsic characterization of when $X$ embeds into $\ell_p \oplus \ell_2$ in terms of weakly null trees in $X$ or, equivalently, in terms of the “infinite asymptotic game” played in $X$. This solves problems concerning small subspaces of $L_p$ originating in the 1970’s. The techniques used were developed over several decades, the most recent being that of weakly null trees developed in the 2000’s.

Authors

Richard Haydon

Brasenose College
Oxford OX1 4AJ
U.K.

Edward Odell

The University of Texas at Austin
Austin, TX 78712

Thomas Schlumprecht

Brasenose College
Oxford OX1 4AJ
U.K.

Current address:

Department of Mathematics
Texas A&M University
College Station, TX 77843