Exercise 18, § 1

Let E be a real normed space, of dimension $\geq 3$. Suppose that there exists a bijective decreasing mapping $\omega$ of 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) Prove that there exists a linear mapping $u$ of E onto its dual $E'$ uniquely determined up to a scalar factor and such that $u(M) = (\omega(M))^\circ$ for every $M \in \mathfrak{M}$. (By considering the case where M has dimension 1, apply the fundamental theorem of projective geometry (A, II, p. 203, exerc. 16), noting that the only automorphism of the field $\mathbf{R}$ is the identity (TG, IV, p. 52, exerc. 3 of § 3).) (b) If one puts $\langle x|y \rangle = \langle x, u(y) \rangle$, prove that $\langle x|x \rangle \neq 0$ for every $x \neq 0$ 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 couple of points $x, y$ of E (consider a number $\lambda \in \mathbf{R}$ such that $\langle \lambda x + y|x \rangle = 0$). (c) Prove that $\langle x|x \rangle$ keeps a constant sign in the set of all $x \neq 0$; replacing $u$ by $-u$ if necessary, one can therefore suppose 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. Prove that the topology $\mathscr{T}$ on E, defined by the norm $\langle x|x \rangle^{1/2}$, is finer than the topology $\mathscr{T}_0$ (notice that the dual of E for $\mathscr{T}$ contains the dual $E'$ of E for $\mathscr{T}_0$). (e) Prove that $u$ is a continuous mapping of 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 $\mathscr{T}_0$, $u$ is continuous for $\mathscr{T}_0$ and for the strong topology $\beta(E', E)$ (notice that $u$ transforms every bounded set for $\sigma(E, E')$ into a bounded set for $\sigma(E', E)$). Deduce that the topologies $\mathscr{T}$ and $\mathscr{T}_0$ are then identical, and that $\omega(M)$ is the orthogonal supplement of M for the Hilbert space structure defined on E by the form $\langle x|y \rangle$ (cf. I, p. 17, th. 1). (f) Prove that in the space $\ell^1(\mathbf{N})$, endowed with the norm induced by that of $\ell^\infty(\mathbf{N})$, there exists a bijective mapping $M \mapsto \omega(M)$ of $\mathfrak{M}$ onto itself, having the properties stated above (IV, p. 47, exerc. 1).