Exercise 19, § 1

Let $E$ be an infinite dimensional complex normed space. Suppose that there exists a bijective mapping $\omega$ from the set $\mathfrak{M}$ of closed vector subspaces of $E$ onto itself, having the properties listed in exerc. 18. a) Show that there exists a semi-linear mapping $u$ from $E$ onto its dual $E'$ (for the automorphism $\xi \mapsto \overline{\xi}$ of $\mathbf{C}$) defined up to a scalar factor and such that $u(M) = (\omega(M))^{\circ}$ for every $M \in \mathfrak{M}$. (Proceed as in exerc. 18; using IV, p. 65, exerc. 16, show that $u$ is a semi-linear mapping relative to the identity automorphism of $\mathbf{C}$ or to the automorphism $\xi \mapsto \overline{\xi}$; finally prove that the first case cannot occur since $\langle x, u(x) \rangle \neq 0$ for $x \neq 0$.) b) If we put $\langle y|x \rangle = \langle x, u(y) \rangle$, show that $\langle x|y \rangle = \langle y|x \rangle$ and that $\langle x|x \rangle$ has the same sign on the set of all $x \neq 0$ (same method as in exerc. 18). c) Finally show that the topology defined by the norm $\langle x|x \rangle^{1/2}$ is finer than the initial topology $\mathscr{T}_0$ on $E$, and that these two topologies are identical when $E$ is complete for $\mathscr{T}_0$; in the latter case, $\omega(M)$ is the orthogonal complement of $M$ in the hilbertian space $E$.