Exercise 25, § 1

a) Let $E$ be a real normed space and $x' \neq 0$ a vector of the dual $E'$ of $E$. Now prove that in order that every vector $y$ of the hyperplane ${x'}^{-1}(0)$ 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 every $x \neq 0$ in $E$, there exists a homogeneous closed hyperplane $H$ of $E$ such that every vector $y \in H$ is quasi-normal to $x$. c) If $x, y$ are two points of $E$ and $x \neq 0$, there exists a scalar $\alpha$ such that $\alpha x + y$ is quasi-normal to $x$.