Linear and multilinear algebra on diffeological vector spaces

This paper deals with some basic constructions of linear and multilinear algebra on finite-dimensional diffeological vector spaces. After verifying that some standard constructions carry over to the diffeological setting, 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 (associativity, distributivity, commutativity with taking), 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). After recalling that a pseudo-metric determines the natural pairing map onto the diffeological dual, which admits a well-defined left inverse for any fixed choice of a basis, we discuss decompositions of vector spaces into a smooth direct sum of the maximal isotropic subspace and a characteristic subspace; we show that, although such a decomposition can be described on the basis of a pseudo-metric, it 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.