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I'm currently reading Halmos's book "Finite dimensional vector spaces" and I find it excellent. However, I'm having some problems with his definition of the tensor product of two vector spaces, and I hope you could help me clear it out. Here's what he writes:

"Definition: The tensor product [tex]U \otimes V [/tex] of two finite dimensional vector spaces U and V (over the same field) is the dual of the vector space of all bilinear forms on [tex] U \oplus V [/tex] . For each pair of vectors x and y, with x in U and y in V, the tensor product [tex] z = x*\otimes y [/tex] of x and y is the element of [tex] U \otimes V [/tex] defined by [tex] z(w) = w(x,y) [/tex] for every bilinear form w."

Just right before that, he talks about that the definition uses reflexivity to obtain the tensor product of U and V. Where? I'm finding this definition somewhat obstruse. And if you guys have the time, I'd love to see an example of how to really get the tensor product of two vector spaces using this definition.

Hope you have the time.