Tensor products of 𝐹-vector spaces

Abstract

Kuranishi [1] has defined the concept of an F-vector space. Each F-vector space determines a pair of numbers (m, p) called the characteristic. The characteristic constitutes a generalization to this class of in general infinite dimensional vector spaces of the notion of dimension for finite dimensional vector spaces. The latter, in fact, correspond to F-vector spaces of characteristic (m, 0) with m equaling the dimension. Isomorphism of F-vector spaces, over the same field of course, is such that they are isomorphic if and only if they have the same characteristic. Except for a very special case, the construction of the space of formal curves, Kuranishi left open the question of forming tensor products of F-vector spaces. It will be shown that the tensor product of two F-vector spaces having characteristics (m, p) and (m', p') respectively exists and is an F-vector space of characteristic (mm', p+p'). It is immediately seen that this generalizes the well-known result for finite dimensional vector spaces. All vector spaces are taken over some infinite field. A PF-vector space H= (H, HM I B (1) h(1)) is a vector space such that (1) HU') and B(1) are vector subspaces for each nonnegative integer 1; (2) H= H(0) D H(1) D . . . D H() D H(1+1) D . . . (3) n, HM)= {O} ; (4) HM') = B (1) E H(1+ ); (5) B(1) has finite dimension dI and h(l)= {h', hl, * *, hll} is an ordered basis for BM. An F-vector space H= (H, HU), B (1) h(1)) is a PF-vector space such that there are integers p and k and a real number ml such that for all sufficiently large I