Is it possible that one inner product is always greater or equal to the other inner product on the same vector space?

Question

For example, there are two inner products $ Q_1$ and $Q_2$ and they are both the inner product on $\mathbb R^2$.

Is it possible to suppose that $ Q_1 \geq Q_2$?

More specifically, for all vectors $v,w\in \mathbb R^2$, is it always true that $Q_1(v,w)\geq Q_2(v,w)$ and $ Q_1(v,w)= Q_2(v,w)\iff$ [$v=w$ and $v=0$]?

Is this possible?

Answer 1

No. If $Q_1(v,w)\geq Q_2(v,w)$ for all $v,w$, you can replace $v$ with $-v$ to get the reverse inequality. In other words $Q_1\geq Q_2$ if and only if $Q_1=Q_2$.