Abstract
LetX be a Banach space. A Banach spaceY is an envelope ofX if (1)Y is finitely representable inX; (2) any Banach spaceZ finitely representable inX and of density character not exceeding that ofY is isometric to a subspace ofY. Lindenstrauss and Pelczynski have asked whether any separable Banach space has a separable envelope. We give a negative answer to this question by showing the existence of a Banach space isomorphic tol 2, which has no separable envelope. A weaker positive result holds: any separable Banach space has an envelope of density character ≦ℵ1 (assuming the continuum hypothesis).
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Stern, J. The problem of envelopes for Banach spaces. Israel J. Math. 24, 1–15 (1976). https://doi.org/10.1007/BF02761425
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DOI: https://doi.org/10.1007/BF02761425