Let $V$ be a Banach space with respect to norms $\|\cdot\|_{1}$ and $\|\cdot\|_2$. There exists $Q>0$ such that $\forall_{v\in V} \|v\|_1\leq Q\|v\|_2$. Show that there exists $P>0$ such that $\forall_{v\in V} \|v\|_2\leq P\|v\|_1$.
I have tried proving it by contradiction, but then I would get a sequence $v_n$ such that $$ \forall_n \|v_n\|_2>n\|v_n\|_1$$ and I could not get anything from it. I guess that I need to find a Cauchy sequence that doesn't converge in order to get a contradiction. Because without assumption of completeness it's obviously false.
Best Answer
The identity map from $(V,\|.\|_2)$ to $ (V,\|.\|_1)$ is continuous and Open Mapping Theorem tells you that the identity map from $(V,\|.\|_1)$ to $ (V,\|.\|_2)$ is continuous. This gives the desired inequality.