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1 definition found
 for tensor product
From The Free On-line Dictionary of Computing (30 December 2018) :

  tensor product
  
      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)
  

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