Journal of Convex Analysis 17 (2010), No. 1, 349--356
Copyright Heldermann Verlag 2010
A Unified Construction Yielding Precisely Hilbert and James Sequences Spaces
Dusan Repovs
Faculty of Mathematics and Physics, University of Ljubljana, P. O. Box 2964, Ljubljana 1001, Slovenia
dusan.repovs@guest.arnes.si
Pavel V. Semenov
Department of Mathematics, Moscow City Pedagogical University, 2-nd Selskokhozyastvennyi pr. 4, Moscow 129226, Russia
pavels@orc.ru
[Abstract-pdf]
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 $l_M$.
Keywords: Hilbert space, Banach space, James sequence space, quasireflexive space, invertible continuous operator, Orlicz function.
MSC: 54C60, 54C65, 41A65; 54C55, 54C20
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