If $V,W$ are infinite-dimensional vector spaces with basis {${v\_i}$} and {${w\_j}$} respectively it holds that $V\otimes W$ has as basis {${v\_i⊗w\_j}$}.


What about the reciprocal? That is: if {${v\_i}$} and {${w\_j}$} are families of vectors in $V$ and $W$ respectively such that the family {${v\_i⊗w\_j}$} is a basis of $V\otimes W$, are {${v\_i}$} and {${w\_j}$} bases of $V$ and $W$ respectively?



Reverse the construction of a basis for a tensor product of vector spaces
I am afraid this question might be downvoted for not being research-level, but let me quickly expand the comment by Padraig Ó Catháin. For a fixed $k$, consider $W\_k=\operatorname{Span}(w\_k)$ and the projection $W\to W\_k$; by composing with the map $(v,\alpha w\_k)\mapsto \alpha v$, you get a bilinear mapping $V\times W \to V \times W\_k \to V$. Hence, by the universal property, a linear onto map $V\otimes W \to V$ showing that $\{v\_i\}$ is a set of generators; linear independence follows from looking at $\{v\_i\otimes w\_k\}$.


