- #1
Newtime
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When I was first introduced to the tensor product, I was actually introduced to a special case: the tensor product of vector spaces over [tex]\mathbb{C}[/tex], which was explained to be as the space of multilinear maps on the cross product of the dual spaces, for example. At the time I wasn't aware this was a special case, but now that I'm working through chapter 4 of Hungerford's Algebra, it's clear.
I feel like I understand the general construction in terms of the universality of the tensor product of two modules in the category of middle linear maps (I think this terminology may be exclusive to Hungerford). My problem is relating this more general construction to the one above. That is, if I replace the ring [tex]R[/tex] with [tex]\mathbb{C}[/tex], and the modules with vector spaces, I don't get what I want. This probably also implies that I don't really understand the tensor product in the more general setting as much as I think I do.
So my question is this: How does one get from the more general construction to the special case mentioned above?
I feel like I understand the general construction in terms of the universality of the tensor product of two modules in the category of middle linear maps (I think this terminology may be exclusive to Hungerford). My problem is relating this more general construction to the one above. That is, if I replace the ring [tex]R[/tex] with [tex]\mathbb{C}[/tex], and the modules with vector spaces, I don't get what I want. This probably also implies that I don't really understand the tensor product in the more general setting as much as I think I do.
So my question is this: How does one get from the more general construction to the special case mentioned above?