I was trying to prove that a certain relation ($\leq$) in a real vector space is a partial order, and I lack the antisymmetric property. However, I have been able to prove the following statement:
If $-\alpha I \leq W \leq \alpha I$ for every $\alpha > 0$ then $W=0$.
Is that condition equivalent to the antisymmetric property? If so, does anyone know how to prove that equivalence or know of any reference where it is done? Thank you.