[Math] equivalent norms in Banach spaces of infinite dimension

Question

Suppose $ X $ is a Banach space with respect to two different norms, $ \|\cdot\|_1 \mathrm{ e } \|\cdot\|_2 $. Suppose there is a constant $ K > 0 $ such that
$$ \forall x \in X, \|x\|_1 \leq K\|x\|_2 .$$
show then that these two norms are equivalent

Answer 1

This is a variant of the open mapping theorem. If we consider the identity map $i$ on $X$ as a linear mapping from $(X, \|\cdot\|_2)$ to $(X, \|\cdot\|_1)$, then your condition says that $i$ is continuous. Then by the theorem $i$ is open and hence a homeomorphism, so its inverse is also continuous and the norms are equivalent.