# 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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topBossard, 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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- [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.
- [L2] Lancien G. , Dentability indices and locally uniformly convex renormings, Rocky Mountain J. Math. 23 (1993), 635-647. Zbl0801.46010
- [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
- [Z] Zippin M., Banach spaces with separable duals, Trans. Amer. Math. Soc. 310 (1988), 371-379. Zbl0706.46015

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