This is an announcement for the paper “On a 'philosophical' question about Banach envelopes” by Fernando Albiachttps://arxiv.org/search/math?searchtype=author&query=Albiac%2C+F, Jose L. Ansorenahttps://arxiv.org/search/math?searchtype=author&query=Ansorena%2C+J+L, Przemyslaw Wojtaszczykhttps://arxiv.org/search/math?searchtype=author&query=Wojtaszczyk%2C+P.

Abstract: We show how to construct nonlocally convex quasi-Banach spaces $X$ whose dual separates the points of a dense subspace of $X$ but does not separate the points of $X$. Our examples connect with a question raised by Pietsch [About the Banach envelope of $l_{1,\infty}$, Rev. Mat. Complut. 22 (2009), 209-226] and shed light into the unexplored class of quasi-Banach spaces with nontrivial dual which do not have sufficiently many functionals to separate the points of the space.

https://arxiv.org/abs/2003.07742