TITLE: equivalent norms in Banach spaces of infinite dimension
QUESTION [13 upvotes]: 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

REPLY [2 votes]: Use the open mapping principle: the identity map $F: (X, \Vert \cdot \Vert_2) \to (X, \Vert \cdot \Vert_1)$ is linear and continuous (thanks to hypothesis). Its rank is of the second category, since $X$ is Banach (in particular, a complete metric space under the metric induced by $\Vert \cdot \Vert_1$). 
By open mapping, it follows the this is an open map, i.e. an homeomorphism. In particular, the inverse map $F^{-1}: (X, \Vert \cdot \Vert_1) \to (X, \Vert \cdot \Vert_2)$ is still continuous; from this you easily get the other inequality you need in order to prove that the norms are equivalent.