do you know any example which is not lattice norm?

Question

A norm $||.||$ on a Riesz space is said to be lattice norm whenever $|x|\leq |y|$ implies $||x||\leq ||y||$. Do you know any example which is not a lattice norm?

Answer 1

Take the Sobolev space $H^1(0,1)$. For example, you could choose $y \equiv 1$ and $x(t) = \sin(n \, t)$ for $n$ large enough.