Let $H$ be a Hilbert space, equipped with an inner product $(\cdot,\cdot)_1$ and norm $\|\cdot\|_1$ induced by it.
Let $(\cdot,\cdot)_2$ be other inner product on $H$ and $\|\cdot\|_2$ the norm induced by $(\cdot,\cdot)_2$.
Suppose there exists $a>0$ such that $\|x\|_2\leq a\|x\|_1$ for all $x\in H$. Is there any case on which this inequality implies that $(H,\|\cdot\|_2)$ is complete?
In other words, to prove that $(H,\|\cdot\|_2)$ is complete it's enough to show that there exists $a,b>0$ such that $\|x\|_2\leq a\|x\|_1$ and $\|x\|_1\leq b\|x\|_2$ for all $x\in H$ (that is, the norms are equivalents). I would like to know if there any case (maybe when we impose some condition on the inner product) on which $\|x\|_2\leq a\|x\|_1$ ensures the completeness.