tensor product

tensor product

   <mathematics> A function of two {vector spaces}, U and V,
   which returns the space of {linear maps} from V's {dual} to U.

   Tensor product has natural symmetry in interchange of U and V
   and it produces an {associative} "multiplication" on vector
   spaces.

   Wrinting * for tensor product, we can map UxV to U*V via:
   (u,v) maps to that linear map which takes any w in V's dual to
   u times w's action on v.  We call this linear map u*v.  One
   can then show that

   	u * v + u * x = u * (v+x)
   	u * v + t * v = (u+t) * v
   and
   	hu * v = h(u * v) = u * hv

   ie, the mapping respects {linearity}: whence any {bilinear
   map} from UxV (to wherever) may be factorised via this
   mapping.  This gives us the degree of natural symmetry in
   swapping U and V.  By rolling it up to multilinear maps from
   products of several vector spaces, we can get to the natural
   associative "multiplication" on vector spaces.

   When all the vector spaces are the same, permutation of the
   factors doesn't change the space and so constitutes an
   automorphism.  These permutation-induced iso-auto-morphisms
   form a {group} which is a {model} of the group of
   permutations.

   (1996-09-27)