Exercise 25, § 1

a) Let E be a real normed space and $x' \neq 0$ a vector in the dual $E'$ of E. Show that for every vector $y$ in the hyperplane ${x'}^{-1}(0)$ to be quasi-normal to $x$ (exerc. 24), it is necessary and sufficient that $\langle x, x' \rangle = \|x\| \cdot \|x'\|$. b) Deduce from a) that for all $x \neq 0$ in E, there exists a closed homogeneous hyperplane H of E such that every vector $y \in H$ is quasi-normal to $x$. c) If $x, y$ are two points in E and $x \neq 0$, then there exists a scalar $\alpha$ such that $\alpha x + y$ is quasi-normal to $x$.