Relation With The Dual Space
In the discussion on the universal property, replacing Z by the underlying scalar field of V and W yields that the space (V ⊗ W)* (the dual space of V ⊗ W, containing all linear functionals on that space) is naturally identified with the space of all bilinear functionals on V × W In other words, every bilinear functional is a functional on the tensor product, and vice versa.
Whenever V and W are finite dimensional, there is a natural isomorphism between V* ⊗ W* and (V ⊗ W)*, whereas for vector spaces of arbitrary dimension we only have an inclusion V* ⊗ W* ⊂ (V ⊗ W)*. So, the tensors of the linear functionals are bilinear functionals. This gives us a new way to look at the space of bilinear functionals, as a tensor product itself.
Read more about this topic: Tensor Product
Famous quotes containing the words relation, dual and/or space:
“You know there are no secrets in America. It’s quite different in England, where people think of a secret as a shared relation between two people.”
—W.H. (Wystan Hugh)
“Thee for my recitative,
Thee in the driving storm even as now, the snow, the winter-day
declining,
Thee in thy panoply, thy measur’d dual throbbing and thy beat
convulsive,
Thy black cylindric body, golden brass and silvery steel,”
—Walt Whitman (1819–1892)
“from above, thin squeaks of radio static,
The captured fume of space foams in our ears—”
—Hart Crane (1899–1932)