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.
Keywords
diffeological vector space, homological algebra, fine
diffeology, projective diffeological vector space, smooth
linear functionals
