Exercise 21, § 1

Let $E$ be a finite dimensional complex vector space, and let $\phi$ be a bijective semi-linear (relative to the automorphism $\xi \mapsto \overline{\xi}$ of $\mathbf{C}$) mapping from $E$ onto its dual $E^*$. Let $\|x\|$ be a norm on $E$ such that, for all $x \in E$, we have $|\langle x, \phi(x) \rangle| = \|x\| \cdot \|\phi(x)\|$. Show that $(y, x) \mapsto \langle x, \phi(y) \rangle$ is, up to a constant factor, a positive separating hermitian form and that $\langle x, \phi(x) \rangle = \gamma \|x\|^2$ ($\gamma$ constant). (Argue as in exerc. 20.)