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:
“In relation to God, we are like a thief who has burgled the house of a kindly householder and been allowed to keep some of the gold. From the point of view of the lawful owner this gold is a gift; From the point of view of the burglar it is a theft. He must go and give it back. It is the same with our existence. We have stolen a little of God’s being to make it ours. God has made us a gift of it. But we have stolen it. We must return it.”
—Simone Weil (1909–1943)
“Thee for my recitative,
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declining,
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—Walt Whitman (1819–1892)
“For good teaching rests neither in accumulating a shelfful of knowledge nor in developing a repertoire of skills. In the end, good teaching lies in a willingness to attend and care for what happens in our students, ourselves, and the space between us. Good teaching is a certain kind of stance, I think. It is a stance of receptivity, of attunement, of listening.”
—Laurent A. Daloz (20th century)