  We study the relationship between many natural conditions that one can put on
a diffeological vector space: being fine or projective, having enough smooth
(or smooth linear) functionals to separate points, having a diffeology
determined by the smooth linear functionals, having fine finite-dimensional
subspaces, and having a Hausdorff underlying topology. Our main result is that
the majority of the conditions fit into a total order. We also give many
examples in order to show which implications do not hold, and use our results
to study the homological algebra of diffeological vector spaces.
