This question is inspired by the MO question Image of $L^1$ under the Fourier transform, but I think it might be much easier so I am posting it here instead.
Let $(X, \|\cdot\|)$ be a separable Banach space, and $E$ a linear subspace.
1. Suppose $E$ is Borel. What can be the complexity of $E$, in the sense of the Borel hierarchy? (For example, as shown here, it cannot be properly $G_\delta$.) Can $E$ have arbitrarily high Borel rank?
Let us say $E$ is Banachable if there is another norm $\|\cdot\|'$ on $E$, stronger than $\|\cdot\|$, under which $E$ is separable Banach. Or equivalently, $E$ is Banachable iff there exists a separable Banach space $Y$ and a continuous injective linear map $T : Y \to X$ whose image is $E$. It is clear that every Banachable subspace is Borel.
2. Is every Borel subspace Banachable?
3. If not, what can be the complexity of a Banachable subspace?
4. Do these answers change if I replace "Banach" by "Hilbert" throughout?
One could also ask the analogous questions for Polish groups.