equivalence of normed spaces and separability

Question

The question can be trivial, but I can not figure it out myself.

Assume we have that $X$ is a normed space, with norm $\| {\cdot}\|_{1}$ and topology generated by the norm. Let us consider another norm, let's say $\|{\cdot}\|_{2}$, which is equivalent to $\|{\cdot}\|_{1}$.

Then, if the space $(X, \| {\cdot}\|_{1})$ is separable, does it follows that the space $(X, \| {\cdot}\|_{2})$ is also separable?

Answer 1

Yes, because $\operatorname{id}\colon(X,\|\cdot\|_1)\longrightarrow(X,\|\cdot\|_2)$ is a homeomorphism, and therefore the image of a dense subset is also a dense subset.