## 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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