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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

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

### Tom 168 / 2005

Studia Mathematica 168 (2005), 203-216 MSC: Primary 46B03; Secondary 03A15. DOI: 10.4064/sm168-3-2

#### Streszczenie

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.

#### Autorzy

• Valentin FerencziEquipe d'Analyse
Université Paris 6
Couloir 46-0, Boîte 186
4, place Jussieu
75252 Paris Cedex 05, France
e-mail
• Christian RosendalEquipe 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