Function spaces not containing $\ell_{1}$

TL;DR

This paper introduces a class of function spaces, including the James space, showing they are separable with non-separable duals and do not contain an isomorphic copy of , with proofs based on topological methods.

Contribution

It defines the space $JF_{X}(\u03a9)$ for reflexive Banach spaces with symmetric bases and proves these spaces do not contain , expanding understanding of their structure.

Findings

Spaces are separable with non-separable duals.

Spaces do not contain an isomorphic  copy.

Includes the classical James function space as a special case.

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 $JF_{X}(\Omega)$ 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 $JF_{X}(\Omega)$ does not contain an isomorphic copy of $\ell_{1}$. We also investigate the structure of these spaces and their duals.