On the number of non-isomorphic subspaces of a Banach space

Valentin Ferenczi ¹   ; Christian Rosendal ²

¹ Equipe d'Analyse Université Paris 6 Couloir 46-0, Boîte 186 4, place Jussieu 75252 Paris Cedex 05, France
² Equipe d'Analyse Université Paris 6 Couloir 46-0, Boîte 186 4, place Jussieu 75252 Paris Cedex 05, France and Mathematics 253-37 California Institute of Technology Pasadena, CA 91125, U.S.A.

Studia Mathematica, Tome 168 (2005) no. 3, pp. 203-216

Résumé

We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let $\frak X$ be a Banach space with an unconditional basis $(e_i)_{i \in {\mathbb N}}$; then either there exists a perfect set $ P$ of infinite subsets of ${\mathbb N}$ such that for any two distinct $A,B \in P$, $[e_i]_{i \in A} \ncong [e_i]_{i \in B}$, or for a residual set of infinite subsets $A$ of ${\mathbb N}$, $[e_i]_{i \in A}$ is isomorphic to $\frak X$, and in that case, $\frak X$ is isomorphic to its square, to its hyperplanes, uniformly isomorphic to ${\frak X} \oplus [e_i]_{i \in D}$ for any $D\subset {\mathbb N}$, and isomorphic to a denumerable Schauder decomposition into uniformly isomorphic copies of itself.

Détail

DOI : 10.4064/sm168-3-2

Keywords: study number non isomorphic subspaces given banach space main result following frak banach space unconditional basis mathbb either there exists perfect set infinite subsets mathbb distinct ncong residual set infinite subsets mathbb isomorphic frak frak isomorphic its square its hyperplanes uniformly isomorphic frak oplus subset mathbb isomorphic denumerable schauder decomposition uniformly isomorphic copies itself