Abstract
We show that for every weakly compact subset K of C[0, 1] with finite Cantor–Bendixson rank, there is a reflexive Banach lattice E and an operator \(T:E\rightarrow C[0,1]\) such that \(K\subseteq T(B_E)\). On the other hand, we exhibit an example of a weakly compact set of C[0, 1] homeomorphic to \(\omega ^\omega +1\) for which such T and E cannot exist. This answers a question of M. Talagrand in the 80’s.
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Support of the Ministerio de Economía y Competitividad under Grant MTM2012-31286 (Spain) is gratefully acknowledged. J. Lopez-Abad has been partially supported by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO). P. Tradacete also supported by Grants MTM2013-40985 and Grupo UCM 910346.
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Lopez-Abad, J., Tradacete, P. Shellable weakly compact subsets of C[0, 1]. Math. Ann. 367, 1777–1790 (2017). https://doi.org/10.1007/s00208-016-1473-3
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DOI: https://doi.org/10.1007/s00208-016-1473-3