Remark

— Suppose that the duality between F and G is separating in G and that the duality between $F_1$ and $G_1$ is separating in $G_1$. If we identify G and $G_1$ with subspaces of $F^*$ and $F_1^*$ respectively, the conditions a) and b) are equivalent to $^t u(G_1) \subset G$; v is the restriction of the transpose $^t u$ of u (A, II, § 2.5) to $G_1$.

We say, simply (when there is no chance of confusion) that v is the transpose of u (relative to the duality on the one hand between F and G and on the other hand between $F_1$ and $G_1$) and we again use $^t u$ to denote it.