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We show how to construct nonlocally convex quasi-Banach spaces $X$ whose dualseparates the points of a dense subspace of $X$ but does not separate thepoints of $X$. Our examples connect with a question raised by Pietsch [Aboutthe 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 nontrivialdual which do not have sufficiently many functionals to separate the points ofthe space.

On a 'philosophical' question about Banach envelopes