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Banach envelopes of non-locally convex spaces. (English) Zbl 0577.46016

Let X be a non-locally convex quasi-Banach space with separating dual. We give conditions which imply that \(X^*\) cannot have finite cotype. These conditions imply that most naturally arising spaces X cannot have \(c_ 0\) as a Banach envelope. However, we also construct a space with \(c_ 0\) as its Banach envelope. Finally we show that for any separable Banach space Z there is a subspace X of \(\ell_ p\) \((0<p<1)\) whose Banach envelope is isomorphic to \(\ell_ 1\oplus Z\).

MSC:

46B25 Classical Banach spaces in the general theory
46B20 Geometry and structure of normed linear spaces
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