Let $(X,\|\cdot\|_X)$ be a separable normed space and let $(Y,\|\cdot\|_Y)$ be a normed space. Assume they are both infinite dimensional. Let $$T:(X,\|\cdot\|_X) \longrightarrow (Y,\|\cdot\|_Y)$$ be a isomorphic isometry. What I'm interested to know is if $(Y,\|\cdot\|_Y)$ also will be separable? What I am really interested to know is wheter or not there can exist such a map between $c$ and $l^\infty$, but it doesn't hurt to be a bit more general.
Take $y\in Y$. Then there exist $x \in X$ such that $T(x) = y$. Also there exist a countable dense subset $X'$ of $X$ such that for every $\epsilon > 0$ there exist $x' \in X'$ such that $\| x' - x \|_X < \epsilon$. But since $T$ is an isometry we get
$$\| x' - x \|_X = \| T(x' - x) \|_Y = \| T(x') - y \|_Y < \epsilon$$
But $T(x') \in T(X')$ and $T(X')$ is countable (T is bijective) so therefore $(Y,\|\cdot\|_Y)$ must separable.
Thank you in advance!