I know that in general, if V,W are infinite-dimensional vector spaces with basis {vi} and {wj} respectively it holds that V⊗W has as basis {vi⊗wj}.
My question is: what about the reciprocal? That is: if {vi} and {wj} are families of vectors in V and W respectively such that the family {vi⊗wj} is a basis of V⊗W, are {vi} and {wj} bases of V and W respectively?
My question is: what about the reciprocal? That is: if {vi} and {wj} are families of vectors in V and W respectively such that the family {vi⊗wj} is a basis of V⊗W, are {vi} and {wj} bases of V and W respectively?