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?