I have a normed space $(E,||\cdot||)$ which is homeomorphic (as a topological space) to a Banach space $F$.

Does this imply that $(E,||\cdot||)$ is also a Banach space?

I think I read something like this to be true if $E$ (and therefore also $F$) is separable, but I am not totally sure. So, also this special case would be interesting.