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?