  This paper deals with some basic constructions of linear and multilinear
algebra on finite-dimensional diffeological vector spaces. We consider the
diffeological dual formally checking that the assignment to each space of its
dual defines a covariant functor from the category of finite-dimensional
diffeological vector spaces to the category of standard (that is, carrying the
usual smooth structure) vector spaces. We verify that the diffeological tensor
product enjoys the typical properties of the usual tensor product, after which
we focus on the so-called smooth direct sum decompositions, a phenomenon
exclusive to the diffeological setting. We then consider the so-called
pseudo-metrics (diffeological analogues of scalar products), and discuss the
consequent decompositions of vector spaces into a smooth direct sum of the
maximal isotropic subspace and a characteristic subspace; we show that such a
decomposition actually depends only on the choice of the coordinate system.
Furthermore, we show that, contrary to what was erroneously claimed (by me)
elsewhere, a characteristic subspace is not unique and is not invariant under
the diffeomorphisms of the space on itself. The maximal isotropic subspace is
on the other hand an invariant of the space itself, and this fact allows to
assign to each its well-defined characteristic quotient, obtaining another
functor, this time a contravariant one, to the category of standard vector
spaces. After discussing the diffeological analogues of isometries, we end with
some remarks concerning diffeological algebras and diffeological Clifford
algebras. Perhaps a larger than usual part of the paper recalls statements that
already appear elsewhere, but when this is the case, we try to accompany them
with new proofs and examples.
