# An ordinal version of some applications of the classical interpolation theorem

Fundamenta Mathematicae (1997)

• Volume: 152, Issue: 1, page 55-74
• ISSN: 0016-2736

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## Abstract

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Let E be a Banach space with a separable dual. Zippin’s theorem asserts that E embeds in a Banach space ${E}_{1}$ with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński have shown that E is a quotient of a Banach space ${E}_{2}$ with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we prove that the Szlenk indices of ${E}_{1}$ and ${E}_{2}$ can be controlled by the Szlenk index of E, where the Szlenk index is an ordinal index associated with a separable Banach space which provides a transfinite measure of the separability of the dual space.

## How to cite

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Bossard, Benoît. "An ordinal version of some applications of the classical interpolation theorem." Fundamenta Mathematicae 152.1 (1997): 55-74. <http://eudml.org/doc/212199>.

@article{Bossard1997,
abstract = {Let E be a Banach space with a separable dual. Zippin’s theorem asserts that E embeds in a Banach space $E_1$ with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński have shown that E is a quotient of a Banach space $E_2$ with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we prove that the Szlenk indices of $E_1$ and $E_2$ can be controlled by the Szlenk index of E, where the Szlenk index is an ordinal index associated with a separable Banach space which provides a transfinite measure of the separability of the dual space.},
author = {Bossard, Benoît},
journal = {Fundamenta Mathematicae},
keywords = {Zippin's theorem; shrinking basis; Szlenk indices; separability of the dual space},
language = {eng},
number = {1},
pages = {55-74},
title = {An ordinal version of some applications of the classical interpolation theorem},
url = {http://eudml.org/doc/212199},
volume = {152},
year = {1997},
}

TY - JOUR
AU - Bossard, Benoît
TI - An ordinal version of some applications of the classical interpolation theorem
JO - Fundamenta Mathematicae
PY - 1997
VL - 152
IS - 1
SP - 55
EP - 74
AB - Let E be a Banach space with a separable dual. Zippin’s theorem asserts that E embeds in a Banach space $E_1$ with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński have shown that E is a quotient of a Banach space $E_2$ with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we prove that the Szlenk indices of $E_1$ and $E_2$ can be controlled by the Szlenk index of E, where the Szlenk index is an ordinal index associated with a separable Banach space which provides a transfinite measure of the separability of the dual space.
LA - eng
KW - Zippin's theorem; shrinking basis; Szlenk indices; separability of the dual space
UR - http://eudml.org/doc/212199
ER -

## References

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1. [B1] Bossard B., Mémoire de Thèse, Université Paris VI, 1994.
2. [B2] Bossard B. , Codage des espaces de Banach séparables. Familles analytiques ou coanalytiques d'espaces de Banach, C. R. Acad. Sci. Paris 316 (1993), 1005-1010.
3. [C] s Christensen J.P.R., Topology and Borel Structure, North-Holland Math. Stud. 10, North-Holland, 1974.
4. Deville R., Godefroy G. and Zizler V., Smoothness and Renormings in Banach spaces, Pitman Monographs and Surveys 64, Longman, 1993. Zbl0782.46019
5. Davis W.J., Figiel T., Johnson W.B. and Pełczyński A., Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311-327. Zbl0306.46020
6. Ghoussoub N., Maurey B. and Schachermayer W., Slicings, selections, and their applications, Canad. J. Math. 44 (1992), 483-504. Zbl0780.46008
7. [J-R] Johnson W.B. and Rosenthal H.P., On ω*-basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77-92.
8. [K-L] Kechris A.S. and Louveau A., Descriptive Set Theory and the Structure of Sets of Uniqueness, London Math. Soc. Lecture Note Ser. 128, Cambridge Univ. Press, 1987. Zbl0642.42014
9. [L1] Lancien G., Théorie de l'indice et problèmes de renormage en géométrie des espaces de Banach, Thèse de doctorat de l'Université Paris VI, 1992.
10. [L2] Lancien G. , Dentability indices and locally uniformly convex renormings, Rocky Mountain J. Math. 23 (1993), 635-647. Zbl0801.46010
11. [S] Szlenk W., The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia Math. 30 (1968), 53-61. Zbl0169.15303
12. [Z] Zippin M., Banach spaces with separable duals, Trans. Amer. Math. Soc. 310 (1988), 371-379. Zbl0706.46015

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