Construction of the Tensor Product

[Tac-Tics]

For some reason, tensors seem to be a terribly mysterious topic, mentioned all the time, but rarely explained in clear terms. Whenever I read a paper which uses them, I get the feeling I'm listening to a blind man talk about an elephant. They have to do with multilinear maps. They are a generalization of vectors and matrices. They are ways to build larger spaces out of smaller ones. But never have I seen a straightforward construction of the darned things!

It seems there are a few distinct kinds of objects called tensors. For concreteness's sake, I'm going to ask about the kind used in this paper I've been reading through on Riemannian Geometry:

http://www.maths.lth.se/matematiklu/personal/sigma/Riemann.pdf

Chapter 8, I believe it is, defines the notion of a Riemann manifold out of the concepts described in all previous chapters. However, it does so through the user of a tensor product, which, while the properties are hinted at, no formal definition is ever made. From what I've seen, it seems to work similarly to the Kronecker product I learned about in quantum computation, but, like every other experience I've had with tensors, it's a bit vague in the details.

So how do you construct a tensor product space? It seems that it has something to do with defining an equivalence class and addition and scalar multiplication on ordered pairs and tuples. There's probably a cleaner way to do it using functions or something. But this is my question to the forum. How can you explicitly define the tensor product without resorting to hand-waving like "the simplest object for which the following is true."

I guess I could go further and ask what the general motivation is behind these objects. They seem pretty abstract, and I can't see how they would be so useful in physics unless they oftered leverage over more naive approaches to problems. But what kind of leverage do they offer?

 

[StatusX]

Here's my favorite article describing them:

http://www.dpmms.cam.ac.uk/~wtg10/tensors3.html

 

[MathematicalPhysicist]

There are notes of some professor from Oxford that I used in learning to Analysis on Manifolds, called "Differentiable Manifolds", I think it's rather in depth coverage:
http://people.maths.ox.ac.uk/~hitchin/hitchinnotes/hitchinnotes.html
Down below.

 

[zhentil]

The easiest way to think of tensor products is on a single vector space. Tensors are just multilinear functions acting on some copies of the vector space and its dual. Tensor products are when you take two tensors and "multiply" them together to get a new tensor. The generalization to manifolds is much easier once you have a firm grasp on what a tensor actually is.

Of course, the best place to learn about tensors is Spivak's Calculus on Manifolds.

 

[Tac-Tics]

Here's my favorite article describing them:

http://www.dpmms.cam.ac.uk/~wtg10/tensors3.html

I've seen this article before. It's very... rambling. It's definitely a different perspective than most others I've read, and probably gets to the point of the darned things, but sadly, in a way which is very hard to follow.

After re-reading it, here's what I understood:

Let X and Y be finite-dimensional linear spaces. Let f be a bilinear function XxY->R and u be in X and v be in Y.
Define
[u, v](f) = f(u, v),
(a[u, v])(f) = a f(u, v)
([u, v] + [u', v')](f) = f(u,v) + f(u', v')

So the set X@Y = {[u, v] for all u in X and v in Y} is a linear space. We also have the properties which follow immediately from the definition:
a[u, v] = [au, v] = [u, av]
[u + u', v] = [u, v] + [u, v']
[u, v + v'] = [u, v] + [u', v]

And lastly, it is easy to show that if x_1, x_2, ..., x_n and y_1, y_2, ..., y_m are bases for X and Y respectively, then {[x_i, y_j] forall i and j} is a basis for X@Y. You show that for any u = SUM a_i x_i and v = SUM b_j y_j, then [u, v] = [SUM a_i x_i, SUM b_j y_j] = SUM SUM a_i b_j [x_i, y_j] by the properties above. If we assign c_k = a_i * b_i (for k ranging from 1 to n * m) and e_k = [x_i, y_j], then we have [u, v] = SUM c_k e_k, so e_k (our [x_i, y_j]'s) form a basis of X@Y.

Lastly, the article talks about the associativity of the tensor product, which seems like it should follow almost trivially from the properties.

Is that really all there is to it? If I'm correct here, it seems so much easier than what people make them out to be. (Maybe it's their simplicity that convinces authors to skimp on their explanations of them?)






The tensor product of X and Y (X@Y in the article)

There are notes of some professor from Oxford that I used in learning to Analysis on Manifolds, called "Differentiable Manifolds", I think it's rather in depth coverage:
http://people.maths.ox.ac.uk/~hitchin/hitchinnotes/hitchinnotes.html
Down below.

I will take a look at this as well when I get home.

The easiest way to think of tensor products is on a single vector space. Tensors are just multilinear functions acting on some copies of the vector space and its dual.

Where do dual spaces come into play in this story? One article I found on general relativity gave a rough outline of how they are important. We treat vectors as linear functions from R->R^n. Then, covectors are simply linear maps from R^n->R. You need an inner product which comes from I don't know where and then a vector v has a dual, v*, which is the covector such that <v*|v> = |v||v*|. Vectors and covectors also have differential representations and that representation is important because it shows a symmetry between the two spaces under a change of basis: v' = dx'/dx v for vectors and v*' = dx/dx' v* for covectors. Or something like that? Then, tensors become the multi-linear maps from VxVxV -> V*xV*xV* or something like that.

Of course, the best place to learn about tensors is Spivak's Calculus on Manifolds.

I'm in the market for a good book on manifolds. I'll definitely check this one out.