# Some properties of the Pisier-Zu interpolation spaces

Colloquium Mathematicae (1993)

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

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topSersouri, 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

top- [BL] J. Bergh and J. Löfström, Interpolation Spaces, Grundlehren Math. Wiss. 223, Springer, 1976. Zbl0344.46071
- [B] J. Bourgain, On convergent sequences of continuous functions, Bull. Soc. Math. Belgique 32 (1980), 235-249. Zbl0474.54008
- [E] G. A. Edgar, A long James space, in: Lecture Notes in Math. 794, Springer, 1980, 31-37.
- [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
- [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
- [PX] G. Pisier and Q. Xu, Random series in the real interpolation spaces between the spaces ${v}_{p}$, preprint.
- [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
- [X] Q. Xu, Espaces d’interpolation réels entre les espaces ${v}_{p}$: Propriétés géométriques et applications probabilistes, preprint. Zbl0718.46060

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