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?



Compactly generated Banach spaces
I think not. Suppose X has a countable subset S whose span is dense in X. Enumerate S as a sequence, then by rescaling you can assume the sequence converges to zero in norm. This gives a countable relatively compact set whose linear span is dense in X.


