The problem of envelopes for Banach spaces |
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Authors: | Jacques Stern |
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Institution: | (1) Université De Paris VII, Paris, France |
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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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Keywords: | |
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