A unified construction yielding precisely Hilbert and James sequences spaces

TL;DR

This paper introduces a unified construction for Banach spaces that precisely characterizes Hilbert and James sequence spaces, revealing a clear dichotomy based on the defining vector.

Contribution

It provides a unified framework for constructing Hilbert and James spaces, showing that all such spaces are either isomorphic to l2 or the James space, depending on the defining vector.

Findings

J(e) is isomorphic to l2 if the sum of e's components is non-zero.

J(e) is isomorphic to the James space J if the sum of e's components is zero.

The dichotomy extends to separable Orlicz sequence spaces.

Abstract

Following James' approach, we shall define the Banach space $J(e)$ for each vector $e=(e_1,e_2,...,e_d) \in \Bbb{R}^d$ with $ e_1 \ne 0$. The construction immediately implies that J(1) coincides with the Hilbert space $i_2$ and that $J(1;-1)$ coincides with the celebrated quasireflexive James space $J$. The results of this paper show that, up to an isomorphism, there are only the following two possibilities: (i) either $J(e)$ is isomorphic to $l_2$, if $e_1+e_2+...+e_d\ne 0$ (ii) or $J(e)$ is isomorphic to $J$. Such a dichotomy also holds for every separable Orlicz sequence space $l_M$.