Many books on algebra say that any arbitrary element in a tensor product is a finite sum of pure tensors, however, I cannot clearly see why this is true. Actually, if both of the spaces $U$ and $V$ are finidimensional, then $U \otimes V$ is finidimensional as well. Then it is really true: any element $x \in U \otimes V$ can be expressed as a finite linear combination of basis elements, and so it can be expressed as a finite sum of pure tensors:
$$
x = \sum_{i=1}^{n} \sum_{j=1}^{m} c_{ij} (u_i \otimes v_j) = \sum_{i=1}^{n} \sum_{j=1}^{m} ((c_{ij} u_i) \otimes v_j).
$$
But why is it true when $U$ and $V$ are not finidimensional? I clearly do not understand this, please give me a hint.
Best Answer
I think your doubt can be expressed just the same as "Why in an infinite dimensional vector space any element in it is a finite linear combination of a (finite, of course) subset of a basis of the space?" .
This is so just definition based on the axioms of vector space (or module, or algebra or whatever): we know how to do finite sums, we do not how to do infinite sums unless some further structure kicks in, say a normed space.