Exercise 1, § 2

Let E be a Riesz space, $x$ an element $> 0$ of E. If there exists a relatively bounded linear form $L$ on E such that $L(x) \neq 0$, show that the set of $nx$ ($n$ an integer $> 0$) cannot be bounded above in E. In particular, show that there does not exist any relatively bounded linear form on the Riesz space E defined in Exer. 5 b) of §1.