Exercise 18, § 1

Let E be a real normed space of dimension $\geq 3$. Suppose that there exists a decreasing bijective mapping $\omega$ from the set $\mathfrak{M}$ of closed vector subspaces of E onto itself, such that $\omega(\omega(M)) = M$ and $M \cap \omega(M) = \{0\}$ for every $M \in \mathfrak{M}$. a) Show that there exists a linear mapping $u$ from E onto its dual $E'$ defined upto a scalar factor and such that $u(M) = (\omega(M))^\circ$ for all $M \in \mathfrak{M}$. (Considering the case where M is 1 dimensional, apply the fundamental th. of projective geometry (A, II, § 9, exerc. 16) by observing that the only automorphism of the field $\mathbf{R}$ is the identity mapping (GT, IV, § 3, exerc. 3). b) If we put $\langle x|y \rangle = \langle x, u(y) \rangle$, show that $\langle x|x \rangle \neq 0$ for all $x \neq 0$ and that the relations $\langle x|y \rangle = 0$ and $\langle y|x \rangle = 0$ are equivalent. Deduce that $\langle y|x \rangle = \langle x|y \rangle$ for every pair of points $x, y$ of E (consider a number $\lambda \in \mathbf{R}$ such that $\langle \lambda x + y|x \rangle = 0$). c) Show that $\langle x|x \rangle$ has the same sign on the set of all $x \neq 0$; replacing $u$ by $-u$ if necessary, we can then assume that $\langle x|y \rangle$ is a positive, separating symmetric bilinear form on $E \times E$. d) Let $\mathscr{T}_0$ be the initial topology of E. Show that the topology $\mathscr{T}$ on E, defined by the norm $\langle x|x \rangle^{1/2}$ is finer than the topology $\mathscr{T}_0$ (observe that the dual of E for $\mathscr{T}$ contains the dual $E'$ of E for $\mathscr{T}_0$). e) Show that $u$ is a continuous mapping from E onto its dual $E'$, for the topologies $\sigma(E, E')$ and $\sigma(E', E)$. Deduce that if E is complete for the initial topology $\mathscr{T}_0$, then $u$ is continuous for $\mathscr{T}_0$ and for the strong topology $\beta(E', E)$ (observe that $u$ transforms every set bounded for $\sigma(E, E')$ into a bounded set for $\sigma(E', E)$). Deduce that then the topologies $\mathscr{T}$ and $\mathscr{T}_0$ are identical, and $\omega(M)$ is the orthogonal complement of $M$ for the hilbertian space structure defined on $E$ by the form $\langle x|y \rangle$ (cf. I, p. 17, th. 1). f) Show that in the space $\ell^1(\mathbf{N})$, with the norm induced by that of $\ell^\infty(\mathbf{N})$ assigned to it, there exists a bijective mapping $M \mapsto \omega(M)$ from $\mathfrak{M}$ onto itself, having the properties mentioned above (IV, p. 47, exerc. 1).