Exercise 21, § 7

With the notation of Theorem 8 of no. 6, let $u$ denote the linear mapping $x \mapsto (\langle x, x_i^* \rangle)$ of $E$ into $K_s^I$ and $y_0 = (\eta_i)$. $K_d^{(I)}$ is identified with a subspace of the dual of $K_s^I$ under the canonical mapping of $K_d^{(I)}$ into its bidual; then show that if $N'$ is the kernel of ${}^t u$, the condition of Theorem 3 (no. 5) expresses the fact that $y_0$ is orthogonal to the intersection of $N'$ and $K_d^{(I)}$. When $I$ is infinite and the system (21) (no. 5) is of finite rank, show that this intersection is distinct from $N'$ (note that $N'$ is then of finite codimension).