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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Banach spaces of polynomials without copies of $ l\sp 1$


Author: Manuel Valdivia
Journal: Proc. Amer. Math. Soc. 123 (1995), 3143-3150
MSC: Primary 46G20; Secondary 46B20
MathSciNet review: 1273528
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Abstract: Let X be a Banach space. For a positive integer m, let $ {\mathcal{P}_{{w^ \ast }}}{(^m}{X^ \ast })$ denote the Banach space formed by all m-homogeneous polynomials defined on $ {X^ \ast }$ whose restrictions to the closed unit ball $ B({X^ \ast })$ of $ {X^ \ast }$ are continuous for the weak-star topology. For each one of such polynomials, its norm will be the supremum of the absolute value in $ B({X^ \ast })$. In this paper the bidual of $ {\mathcal{P}_{{w^ \ast }}}{(^m}{X^ \ast })$ is constructed when this space does not contain a copy of $ {l^1}$. It is also shown that, whenever X is an Asplund space, $ {\mathcal{P}_{{w^ \ast }}}{(^m}{X^ \ast })$ is also Asplund.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1273528-0
PII: S 0002-9939(1995)1273528-0
Article copyright: © Copyright 1995 American Mathematical Society