A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces

Benoit Bossard

Fundamenta Mathematicae (2002)

  • Volume: 172, Issue: 2, page 117-152
  • ISSN: 0016-2736

Abstract

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When the set of closed subspaces of C(Δ), where Δ is the Cantor set, is equipped with the standard Effros-Borel structure, the graph of the basic relations between Banach spaces (isomorphism, being isomorphic to a subspace, quotient, direct sum,...) is analytic non-Borel. Many natural families of Banach spaces (such as reflexive spaces, spaces not containing ℓ₁(ω),...) are coanalytic non-Borel. Some natural ranks (rank of embedding, Szlenk indices) are shown to be coanalytic ranks. Applications are given to universality questions. Analogous results are shown for basic sequences modulo equivalence.

How to cite

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Benoit Bossard. "A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces." Fundamenta Mathematicae 172.2 (2002): 117-152. <http://eudml.org/doc/286352>.

@article{BenoitBossard2002,
abstract = {When the set of closed subspaces of C(Δ), where Δ is the Cantor set, is equipped with the standard Effros-Borel structure, the graph of the basic relations between Banach spaces (isomorphism, being isomorphic to a subspace, quotient, direct sum,...) is analytic non-Borel. Many natural families of Banach spaces (such as reflexive spaces, spaces not containing ℓ₁(ω),...) are coanalytic non-Borel. Some natural ranks (rank of embedding, Szlenk indices) are shown to be coanalytic ranks. Applications are given to universality questions. Analogous results are shown for basic sequences modulo equivalence.},
author = {Benoit Bossard},
journal = {Fundamenta Mathematicae},
keywords = {codings; analytic relations; coanalytic sets; families of Banach spaces; Szlenk index},
language = {eng},
number = {2},
pages = {117-152},
title = {A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces},
url = {http://eudml.org/doc/286352},
volume = {172},
year = {2002},
}

TY - JOUR
AU - Benoit Bossard
TI - A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces
JO - Fundamenta Mathematicae
PY - 2002
VL - 172
IS - 2
SP - 117
EP - 152
AB - When the set of closed subspaces of C(Δ), where Δ is the Cantor set, is equipped with the standard Effros-Borel structure, the graph of the basic relations between Banach spaces (isomorphism, being isomorphic to a subspace, quotient, direct sum,...) is analytic non-Borel. Many natural families of Banach spaces (such as reflexive spaces, spaces not containing ℓ₁(ω),...) are coanalytic non-Borel. Some natural ranks (rank of embedding, Szlenk indices) are shown to be coanalytic ranks. Applications are given to universality questions. Analogous results are shown for basic sequences modulo equivalence.
LA - eng
KW - codings; analytic relations; coanalytic sets; families of Banach spaces; Szlenk index
UR - http://eudml.org/doc/286352
ER -

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