Exercise 16, § 4

a) Let E and F be two infinite dimensional complex normed spaces, and $u$ a bijective semilinear mapping from E onto F, with respect to an automorphism $\sigma$ of $\mathbf{C}$, which transforms every closed hyperplane of E into a closed hyperplane of F. Show that the automorphism $\sigma$ of $\mathbf{C}$ is necessarily continuous (and consequently, is either the identity or the automorphism $\xi \mapsto \overline{\xi}$). (Argue by reductio ad absurdum: let $(x_n)$ be a sequence of points in E satisfying the condition of exerc. 15, a) and let $(\lambda_n)$ be a bounded sequence of complex numbers such that $|\lambda_n^\sigma| \geq n.\ \|u(x_n)\|$ for all $n$; if $x' \in E'$ is such that $\langle x_n, x' \rangle = \lambda_n$ for all $n$, consider the image under $u$ of the closed hyperplane of all $x \in E$ such that $\langle x, x' \rangle = 1$ and derive a contradiction.) b) Deduce from a) that $u$ is a continuous semi-linear mapping from E into F (IV, p. 7, corollary). c) Let $\sigma$ be a discontinuous automorphism of the field $\mathbf{C}$. Show that the bijection $(\xi_n) \mapsto (\xi_n^\sigma)$ from $\mathbf{C}^N$ onto itself transforms every closed hyperplane into a closed hyperplane (cf. IV, p. 14, prop. 15).