The optimality of James’s distortion theorems

Abstract:

A renorming of $\ell _{1}$, explored here in detail, shows that the copies of $\ell _{1}$ produced in the proof of the Kadec-Pełczyński theorem inside nonreflexive subspaces of $L_{1}[0,1]$ cannot be produced inside general nonreflexive spaces that contain copies of $\ell _{1}$. Put differently, James’s distortion theorem producing one-plus-epsilon-isomorphic copies of $\ell _{1}$ inside any isomorphic copy of $\ell _{1}$ is, in a certain sense, optimal. A similar renorming of $c_{0}$ shows that James’s distortion theorem for $c_{0}$ is likewise optimal.