Some properties of the Pisier-Zu interpolation spaces

A. Sersouri

Colloquium Mathematicae (1993)

  • Volume: 65, Issue: 1, page 43-50
  • ISSN: 0010-1354

Abstract

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For a closed subset I of the interval [0,1] we let A(I) = [v1(I),C(I)](1/2)2. We show that A(I) is isometric to a 1-complemented subspace of A(0,1), and that the Szlenk index of A(I) is larger than the Cantor index of I. We also investigate, for ordinals η < ω1, the bases structures of A(η), A*(η), and A * ( η ) [the isometric predual of A(η)]. All the results of this paper extend, with obvious changes in the proofs, to the interpolation spaces [ v 1 ( I ) , C ( I ) ] θ q .

How to cite

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Sersouri, A.. "Some properties of the Pisier-Zu interpolation spaces." Colloquium Mathematicae 65.1 (1993): 43-50. <http://eudml.org/doc/210203>.

@article{Sersouri1993,
abstract = {For a closed subset I of the interval [0,1] we let A(I) = [v1(I),C(I)](1/2)2. We show that A(I) is isometric to a 1-complemented subspace of A(0,1), and that the Szlenk index of A(I) is larger than the Cantor index of I. We also investigate, for ordinals η < ω1, the bases structures of A(η), A*(η), and $A_\{*\}(η)$ [the isometric predual of A(η)]. All the results of this paper extend, with obvious changes in the proofs, to the interpolation spaces $[v_1(I),C(I)]_\{θq\}$.},
author = {Sersouri, A.},
journal = {Colloquium Mathematicae},
keywords = {1-complemented subspace; Szlenk index; Cantor index; interpolation spaces},
language = {eng},
number = {1},
pages = {43-50},
title = {Some properties of the Pisier-Zu interpolation spaces},
url = {http://eudml.org/doc/210203},
volume = {65},
year = {1993},
}

TY - JOUR
AU - Sersouri, A.
TI - Some properties of the Pisier-Zu interpolation spaces
JO - Colloquium Mathematicae
PY - 1993
VL - 65
IS - 1
SP - 43
EP - 50
AB - For a closed subset I of the interval [0,1] we let A(I) = [v1(I),C(I)](1/2)2. We show that A(I) is isometric to a 1-complemented subspace of A(0,1), and that the Szlenk index of A(I) is larger than the Cantor index of I. We also investigate, for ordinals η < ω1, the bases structures of A(η), A*(η), and $A_{*}(η)$ [the isometric predual of A(η)]. All the results of this paper extend, with obvious changes in the proofs, to the interpolation spaces $[v_1(I),C(I)]_{θq}$.
LA - eng
KW - 1-complemented subspace; Szlenk index; Cantor index; interpolation spaces
UR - http://eudml.org/doc/210203
ER -

References

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  1. [BL] J. Bergh and J. Löfström, Interpolation Spaces, Grundlehren Math. Wiss. 223, Springer, 1976. Zbl0344.46071
  2. [B] J. Bourgain, On convergent sequences of continuous functions, Bull. Soc. Math. Belgique 32 (1980), 235-249. Zbl0474.54008
  3. [E] G. A. Edgar, A long James space, in: Lecture Notes in Math. 794, Springer, 1980, 31-37. 
  4. [JZ] K. John and V. Zizler, Smoothness and its equivalent in weakly compactly generated Banach spaces, J. Funct. Anal. 15 (1974), 1-15. Zbl0272.46012
  5. [P] G. Pisier, Sur les espaces de Banach qui ne contiennent pas uniformément de l n 1 , C. R. Acad. Sci. Paris 277 (1973), 991-994. Zbl0271.46011
  6. [PX] G. Pisier and Q. Xu, Random series in the real interpolation spaces between the spaces v p , preprint. 
  7. [S] W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia Math. 30 (1968), 53-61. Zbl0169.15303
  8. [X] Q. Xu, Espaces d’interpolation réels entre les espaces v p : Propriétés géométriques et applications probabilistes, preprint. Zbl0718.46060

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