Exercise 19, § 1

Let $E$ be a complex normed space, of infinite dimension. Suppose that there exists a bijective mapping $\omega$ of the set $\mathscr{M}$ of closed vector subspaces of $E$ onto itself, having the properties stated in Exercise 18. (a) Now prove that there exists a semilinear mapping $u$ of $E$ onto its dual $E’$ (for the automorphism $\xi \mapsto \overline{\xi}$ of $\mathbf{C}$) uniquely determined up to a scalar factor such that $u(M) = (\omega(M))^{\circ}$ for every $M \in \mathscr{M}$. (Proceed as in Exercise 18; using IV, p. 65, Exercise 16, prove that $u$ is a semilinear 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, by observing that $\langle x, u(x) \rangle \neq 0$ for $x \neq 0$.) (b) If one sets $\langle y|x \rangle = \langle x, u(y) \rangle$, prove that $\langle x|y \rangle = \overline{\langle y|x \rangle}$ and that $\langle x|x \rangle$ has a constant sign in the set of the $x \neq 0$ (same method as in Exercise 18). (c) Finally prove 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 this last case, $\omega(M)$ is the orthogonal supplement of $M$ in the Hilbert space $E$.