Exercise 21, § 1

Let E be a finite-dimensional complex vector space, φ a semilinear mapping (relative to the automorphism $\xi \mapsto \overline{\xi}$ of $\mathbf{C}$) bijective from E onto its dual E*. Let $\|x\|$ be a norm on E such that, for every $x \in E$, one has $|\langle x, \varphi(x) \rangle| = \|x\| \cdot \|\varphi(x)\|$. Prove that $(y, x) \mapsto \langle x, \varphi(y) \rangle$ is, up to a constant factor, a separating positive hermitian form, and that one has $\langle x, \varphi(x) \rangle = \gamma \|x\|^2$ (γ constant). (Reason as in Exercise 20.)