  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.
