On the number of non-isomorphic subspaces of a Banach space
Volume 168 / 2005
Studia Mathematica 168 (2005), 203-216
MSC: Primary 46B03; Secondary 03A15.
DOI: 10.4064/sm168-3-2
Abstract
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.
