A Unified Construction Yielding Precisely Hilbert and James Sequences Spaces

Following R. C. 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 $l_2$ 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 these two possibilities: (i) $J(e)$ is isomorphic to $l_2$ if $e_1+e_2+...+e_d\ne 0$, and (ii) $J(e)$ is isomorphic to $J$ if $e_1+e_2+...+e_d =0$. Such a dichotomy also holds for every separable Orlicz sequence space $l_M$.