Suppose that $X$ is a Banach space (or more generally, Frechet space) such that $X$ is the closure of the span of a compact (*in the original topology*) subset $K$. Do we know anything "nice" about $X$, from this information alone?

If $X$ is the span of $K$, without needing take a closure, then a Baire category argument shows that $X$ is locally compact, hence finite dimensional. Can something like this be made to happen in the dense case?

More disconcertingly, could all separable Banach spaces be of this form?