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1991 | 100 | 3 | 251-282
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Pseudocomplémentation dans les espaces de Banach

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This paper introduces the following definition: a closed subspace Z of a Banach space E is pseudocomplemented in E if for every linear continuous operator u from Z to Z there is a linear continuous extension ū of u from E to E. For instance, every subspace complemented in E is pseudocomplemented in E. First, the pseudocomplemented hilbertian subspaces of $L¹$ are characterized and, in $L^p$ with p in [1, + ∞[, classes of closed subspaces in which the notions of complementation and pseudocomplementation are equivalent are pointed out. Then, for Banach spaces with the uniform approximation property, Dvoretzky's theorem is strengthened by proving that they contain uniformly pseudocomplemented $ℓ^2_n$'s. Finally, the study of Banach spaces in which every closed subspace is pseudocomplemented is started.
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  • Équipe d'Analyse, U.A. N° 754 AU C.N.R.S., Université Paris VI, Tour 46, 4ème Étage, 4, Place Jussieu, 75 252 Paris, Cedex 05, France
Bibliografia
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  • [DVO] A. Dvoretzky, Some results on convex bodies and Banach spaces, in : Proc. Internat. Sympos. Linear Spaces, Jerusalem 1960, Israel Acad. Sci. Humanities, Jerusalem 1961, 123-160.
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  • [PIS 1] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986.
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  • [PIS 4] G. Pisier, Bases, suites lacunaires dans les espaces $L^r$ d'après Kadec-Pełczyński, Séminaire Maurey-Schwartz, 1972-73, exposé 18, Ecole Polytechnique, Paris.
  • [PIS 5] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, 1989.
  • [PIS 6] G. Pisier, Les inégalités de Khintchine-Kahane d'après C. Borell, Séminaire sur la géométrie des espaces de Banach, 1977-78, n° 7, Ecole Polytechnique, Palaiseau.
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