What follows arises from conversations with A. Kock. Let B be a cartesian closed category and let Vect_R(B) demote the category of R-vector spaces (=modules) in B, where R is a ring in B. The restricted double dual of an object B of B, denoted by D (B) is defined as D (B) = Hom_R(R^B,R) where Hom_R is the object of homomorphisms and R^B has its pointwise vector space structure. Evaluating gives a map k : B -> Hom_R(R^B,B), Now assume that free vector spaces exist, that is, a left adjoint to the forgetful functor Vect_R( B ). The map k above induces a homomorphism h : F(B) -> D (B). We call B linearly finite when h is an isomorphism. If B = Set and R is any field then linear finite is equivalent to finite. This needs decidability to prove. It is even interesting to ask when h is a mono. This situation doesn't cover the motivating example ie. where D (B) is distributions on B, B is a manifold, and the statement that h is an iso is Walbroek's theorem. The problem is we camnot use Vect but some form of complete vector spaces.