A unified construction yielding precisely Hilbert and James sequences spaces

Following James’ approach, we shall define the Banach space $J(e)$ for each vector $e=(e_1,e_2,\mathrm{.}\mathrm{.}\mathrm{.},e_d)\in {\mathbb{R}}^d$ with $e_1\mathrm{\ne }0$. 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 $e_1+e_2+\mathrm{.}\mathrm{.}\mathrm{.}+e_d\mathrm{\ne }0$ (ii) or $J(e)$ is isomorphic to $J$. Such a dichotomy also holds for every separable Orlicz sequence space $l_M$.