I recently stumbled upon following reasoning:
"Let $X$ be a banachspace with respect to the norms $|\cdot |_1$ and $|\cdot |_2$. Define $|x | := |x |_1 + |x |_2$. Clearly $id: (X,|\cdot|) \to (X,|\cdot|_i)$ is continuous and invertible. Since $(X,|\cdot|)$ is also a banachspace, all three norms must be equivalent by the open mapping theorem."
The problem is, I do not see any reason, why $(X,|\cdot|)$ should be a banachspace but I am unable to construct a counterexample and I think such a counterexample would be rather unintuitive.
I would like to know, wheter or not the above statement is true. If not, is there an easy counterexample?