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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some descriptive set-theoretic properties of the isomorphism relation between Banach spaces


Author: Andrzej Komisarski
Journal: Proc. Amer. Math. Soc. 129 (2001), 3085-3090
MSC (2000): Primary 03E15; Secondary 46B03
Published electronically: April 2, 2001
MathSciNet review: 1840115
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Abstract | References | Similar Articles | Additional Information

Abstract:

Consider the space $\mathcal{V} (E)$ of closed linear subspaces of a separable Banach space $E$equipped with the standard Effros Borel structure. The isomorphism relation between Banach spaces being elements of  $\mathcal{V}(E)$ determines a partition of  $\mathcal{V}(E)$. In this note we prove a result describing the complexity of analytic subsets of  $\mathcal{V}(E)$ intersecting a large enough number of the above-mentioned parts of  $\mathcal{V}(E)$.


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  • [B-Pe] C. Bessaga and A. Pełczyński, Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions, Studia Math. 19 (1960), 53–62. MR 0113132 (22 #3971)
  • [Bo] Benoît Bossard, Codages des espaces de Banach séparables. Familles analytiques ou coanalytiques d’espaces de Banach, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 10, 1005–1010 (French, with English and French summaries). MR 1222962 (94g:46022)
  • [CGP] J. Chaber, G. Gruenhage, and R. Pol, On a perfect set theorem of A. H. Stone and N. N. Lusin’s constituents, Fund. Math. 148 (1995), no. 3, 309–318. MR 1367597 (96k:54061)
  • [Ch-P] J. Chaber, R. Pol, On the Cantor-Bendixson derivative, resolvable ranks and perfect set theorems of A. H. Stone, Israel J. Math. 110 (1999), 103-123. CMP 2000:11
  • [Chr] J. P. R. Christensen, Topology and Borel structure, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. Descriptive topology and set theory with applications to functional analysis and measure theory; North-Holland Mathematics Studies, Vol. 10. (Notas de Matemática, No. 51). MR 0348724 (50 #1221)
  • [Ka] V. G. Kanovei, On uncountable sequences of sets determined by sieve operations, Dokl. Akad. Nauk SSSR 257 (1981), 808-812.
  • [Ke] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597 (96e:03057)
  • [Ku] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR 0217751 (36 #840)
  • [K-M] Kazimierz Kuratowski and Andrzej Mostowski, Set theory, Second, completely revised edition, North-Holland Publishing Co., Amsterdam-New York-Oxford; PWN—Polish Scientific Publishers, Warsaw, 1976. With an introduction to descriptive set theory; Translated from the 1966 Polish original; Studies in Logic and the Foundations of Mathematics, Vol. 86. MR 0485384 (58 #5230)
  • [L-T] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, Springer-Verlag, Berlin, 1996.
  • [Se] Zbigniew Semadeni, Banach spaces of continuous functions. Vol. I, PWN—Polish Scientific Publishers, Warsaw, 1971. Monografie Matematyczne, Tom 55. MR 0296671 (45 #5730)

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Additional Information

Andrzej Komisarski
Affiliation: Department of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland
Email: andkom@mimuw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05925-1
PII: S 0002-9939(01)05925-1
Received by editor(s): July 28, 1999
Received by editor(s) in revised form: March 5, 2000
Published electronically: April 2, 2001
Communicated by: Alan Dow
Article copyright: © Copyright 2001 American Mathematical Society