It is true that if the sequences v_{i}' and w_{i}' are each a basis, then every element in [itex]V \otimes W[/itex] can be written uniquely as
[tex]\sum_i \sum_j c_{ij} (v_i' \otimes w_j')[/tex]
But in my previous post I was simply stating something weaker. I was not assuming that v_{i}' and w_{i}' were bases; just they were just sequences of basis vectors. In particular, some basis vectors may be repeated, and others might not appear at all.
