Remark

Let E and F be two complete normed spaces over $\mathbf{R}$ (or over $\mathbf{C}$), and $(\mathbf{x}, \mathbf{y}) \mapsto \langle \mathbf{x}, \mathbf{y} \rangle$ a continuous bilinear form on $\mathrm{E} \times \mathrm{F}$, such that the relation “$\langle \mathbf{x}, \mathbf{y} \rangle = 0$ for all $\mathbf{y} \in \mathrm{F}$” (resp. $\langle \mathbf{x}, \mathbf{y} \rangle = 0$ for all $\mathbf{x} \in \mathrm{E}$”) implies $\mathbf{x} = 0$ (resp. $\mathbf{y} = 0$). Suppose further that for every $t \in \mathrm{J}$ there is a continuous linear map $B(t)$ of F into itself, such that $\langle A(t).\mathbf{x}, \mathbf{y} \rangle + \langle \mathbf{x}, B(t).\mathbf{y} \rangle = 0$ for every $(\mathbf{x}, \mathbf{y}) \in \mathrm{E} \times \mathrm{F}$. In these circumstances one sees as before that for a map $t \mapsto \mathbf{v}(t)$ of J into F, a primitive of a regulated function, to be such that $\langle \mathbf{u}(t), \mathbf{v}(t) \rangle$ is constant for every integral $\mathbf{u}$ of (4), it is necessary and sufficient that $\mathbf{v}$ should be an integral of (16), which again we call the adjoint of (4).