Corollary

Let K be a field, E a right vector space over K, F a left vector space over K, $(M_\alpha)_{\alpha \in A}$ a family of vector subspaces of E and $(N_\beta)_{\beta \in B}$ a family of vector subspaces of F. Then

It is obviously sufficient to prove the particular case

Clearly the left hand side of (25) is contained in the right hand side. To prove the converse, we consider a basis $(f_\lambda)_{\lambda \in L}$ of F. Every element of $E \otimes_K F$ can then be expressed uniquely in the form $\sum_{\lambda \in L} x_\lambda \otimes f_\lambda$, where $x_\lambda \in E$ (§ 3, no. 7, Corollary 1 to Proposition 7); if $E'$ is a vector subspace of E, the relation $\sum_{\lambda \in L} x_\lambda \otimes f_\lambda \in E' \otimes_K F$ is equivalent, by Proposition 14, to $x_\lambda \in E'$ for all $\lambda \in L$. To say that $\sum_{\lambda \in L} x_\lambda \otimes f_\lambda$ belongs to each of the $M_\alpha \otimes_K F$ thus means that for all $\lambda \in L$ and all $\alpha \in A$, $x_\lambda \in M_\alpha$, that is $x_\lambda \in \bigcap_{\alpha \in A} M_\alpha$ for all $\lambda \in L$, which proves that the right hand side of (25) is contained in the left hand side.