  It is known that the only finite-dimensional diffeological vector space that
admits a diffeologically smooth scalar product is the standard space of
appropriate dimension. In this note we consider a way to circumnavigate this
issue, by introducing a notion of pseudo-metric, which, said informally, is the
least-degenerate symmetric bilinear form on a given space. We apply this notion
to make some observation on subspaces which split off as smooth direct summands
(providing examples which illustrate that not all subspaces do), and then to
show that the diffeological dual of a finite-dimensional diffeological vector
space always has the standard diffeology and in particular, any pseudo-metric
on the initial space induces, in the obvious way, a smooth scalar product on
the dual.
