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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

Pavel V. Semenov
Department of Mathematics, Moscow City Pedagogical University, 2-nd Selskokhozyastvennyi pr. 4, Moscow 129226, Russia


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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