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I am reading "Riemannian Geometry" by Gallot. And I am confused with the following definition of tensor product:

Let $E$ and $F$ are two finite dimensional vector spaces, a vector space $E\otimes F$, unique up to isomorphism and such that for any vector space $G$, $L(E\otimes F,G)$ is isomorphic to $L_2(E\times F,G)$ (the vector space of bilinear maps from $E\times F$ to $G$): $E\otimes F$ is the tensor product of $E$ and $F$. Moreover, there exists a bilinear map from $E\times F$ to $E\otimes F$, denoted by $\otimes$, and such that if $e_i$ and $f_j$ are basis for $E$ and $F$, $(e_i\otimes f_j)$ is basis for $E\otimes F$.

I am confused about the words after "moreover". It seems that the words before "moreover" is the definition of tensor product of two vector spaces. But I don't know how to prove the existence of the bilinear map satisfying the basis requirement. It seems that the definition of tensor product on some online sources also includes the existence of the bilinear map. Which one is correct? This is the first time I encounter tensor product. Thanks!

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  • $\begingroup$ See math.stackexchange.com/questions/51155/… for a thorough take on the bilinear form. $\endgroup$ – Autolatry Jan 23 '15 at 15:30
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    $\begingroup$ The basic idea of a tensor product (at least in LA) is that you want to be able to replace multilinear maps from a product of vector spaces into any vector space by a linear map from a single vector space into the same target. The definition you cite is a bit unfortunate in my opinion, since it is only stated in the special case of bilinear maps. They possibly thought this reduces the complexity of the definition, but I'm afraid it does not. If you define the tensor product using the universal property the map they mention is part of the def.. See en.wikipedia.org/wiki/Tensor_product $\endgroup$ – Thomas Jan 23 '15 at 15:33
  • $\begingroup$ @Autolatry sorry. May I know which answer you are referring? I am still confused. $\endgroup$ – John Jan 23 '15 at 15:46
  • $\begingroup$ an intro: juanmarqz.wordpress.com/cucei-maths/… From Gallot, you should keep reading until "It is easier to understand this when the vector spaces E and F ..." $\endgroup$ – janmarqz Jan 23 '15 at 16:04
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The bilinear map you seek is simply the image of the identity under the isomorphism $$L(E \otimes F, E \otimes F) \to L_2(E \times F, E \otimes F)$$ in your definition specialized to $G = E \otimes F$.

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Your question is "Which one is correct?":

  1. defining tensor products from a construction, or
  2. defining tensor products from a universal property like the book you cite

Answer:

Both are correct, but (2) is more conceptual, and perhaps more general and closer to the "truth" or goal of tensor products. Even though your book may not provide an explicit construction, this is usually done after a definition in the style of (2), and will look like any variant of (1) anyways; but the construction is usually forgotten soon afterwards, focusing on more beautiful proofs that rely on the core universal property.

PS: contrary to comment given above, it doesn't really matter if you're restricting to bilinear maps vs multilinear maps at first.

Two respected books for comparing these two styles of definitions are, respectively:

  1. F. Warner, "Foundations of Differentiable Manifolds and Lie Groups"
  2. H. Federer, "Geometric Measure Theory" (which I heard is still an excellent reference for normed tensor theory)
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