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
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