>>12075518An alternative definition is a multilinear map on products of q vector spaces V with field F and p duals of that space and this makes it a (p,q) tensor. Suppose we have a given (0,2) tensor, g. Then, g(u,v) is a number in the field. We can actually turn this into a (1,1) tensor since each fixed u=c defines a function g(c,v)->F, a dual vector in the dual space. In this manner, the vectors of u act as a function on the v and in this type of prescription, it is not a (0,2) tensor anymore but a (1,1) tensor. This duality between the vector and dual vectors is part of the intuitive meaning of this definition.
The third definition is a lot more abstract-it uses the tensor product. In essence, this constructs a kind of 'universal' object in the following sense: if we have a multilinear map V x W into a given space Z, that there is a map from the tensor product to Z, in a manner that given the map from the tensor product to V x W, have factored it. In this sense, the tensor product represents the most universal object in these types of objects under multilinear maps. A tensor is then the tensor product of p vector spaces V and q dual spaces of that.
