Exercise 4, § 2

Denote by K the field $\mathbf{R}$ with the absolute value $p(\xi) = |\xi|^{1/2}$. Let E be the vector space over K of the real valued regulated functions defined over $I = \{0, 1\}$, continuous on the right everywhere and zero at the point 1 ; show that on E the mapping $x \mapsto \|x\| = \int_0^1 |x(t)|^{1/2} dt$ is a norm. Show that for every function $x \geq 0$ in E, there exists in E two functions $x_1 \geq 0$, $x_2 \geq 0$ such that $x = \frac{1}{2}(x_1 + x_2)$ and $\|x_1\| = \|x_2\| = \frac{1}{\sqrt{2}} \|x\|$. Deduce that every continuous linear form on E is identically zero.