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Tensors, tensor products

Дата публикации: 15-10-2026 22:38:16



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  • Level: Undergrad 
  • Thread starter Thread starter wrobel
  • Start date Start date Thursday, 11:45 AM
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Just found a really good lecture on tensors. It starts with the absolute basics and goes all the way up to universal mapping:
I just want to recall the standard realization of the tensor product.

Let ##X, Y## be vector spaces, say over ##\mathbb{R}##. These spaces need not be finite-dimensional.
Let ##T## denote the space of bilinear maps ##b:X\times Y\to \mathbb{R}##, and let ##T^*## stand for the corresponding algebraic dual space. By definition, ##x\otimes y## is the element of ##T^*## defined by ##(x\otimes y)(b):=b(x,y).##
Thus, we have a mapping ##\psi:X\times Y\to T^*## given by ##\psi(x,y)=x\otimes y.##
We define the tensor product of the spaces as ##X\otimes Y:=\text{span}\,\psi(X\times Y)##

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My first gut reaction was, wouldn't you want ##x\otimes y## to be in ##T## not ##T^*##?

Then I realized I flipped your ##T## and ##T^*## in my head as I'm used to attaching a star to the space of maps of vectors.

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