Function spaces not containing ${\ell }_1$

Abstract

For $\Omega$ bounded and open subset of ${\mathbb{R}}^{d_0}$ and $X$ a reflexive Banach space with 1-symmetric basis, the function space $J{F}_X(\Omega )$ is defined. This class of spaces includes the classical James function space. Every member of this class is separable and has non-separable dual. We provide a proof of topological nature that $J{F}_X(\Omega )$ does not contain an isomorphic copy of ${\ell }_1$. We also investigate the structure of these spaces and their duals.