Proposition 1

Let F, G be two real vector spaces and B a bilinear form on $F \times G$. Let

be linear mappings of G in the dual $F^*$ of F and of F in the dual $G^*$ of G, associated respectively to the right and to the left of B (A, IX, § 1.1). Then B puts F and G in a duality separating in G (resp. in F), if and only if $d_B$ (resp. $s_B$) is injective.*

When F and G are put in separating duality by B, we often identify F (resp. G) with a subspace of $G^*$ (resp. $F^*$) by means of $s_B$ (resp. $d_B$). When we consider F (resp. G) as a subspace of $G^*$ (resp. $F^*$) without specifying how this identification is to be made, we are always using the preceding identifications; the bilinear form B is then identified with the restriction to $F \times G$ of the canonical bilinear form :