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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?

flag	asked Feb 21 at 5:29 Iian Smythe 209●6

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As pointed out by Yemon Choi, your condition is equivalent to separability. However, if you weaken it in the natural way by replacing the condition of compactness to weak compactness, you get an interesting class of spaces (called WCG spaces for obvious reasons) which have remarkable properties and have been studied intensively. The initial paper was a seminal one by Amir and Lindenstrauss in the Annals (available on line) and it is an easy task to locate more recent developments. – jbc Feb 21 at 9:03

Yemon Choi · Accepted Answer · 2013-02-21 05:40:13Z UTC

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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.

answered Feb 21 at 5:40

Yemon Choi
8,436●1●19●44

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And thus we do know something "nice" about $X$: it is separable! – Nik Weaver Feb 21 at 5:56

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@Nik: Isn't the Argyros-Haydon space, being an isomorphic predual of $\ell^1$, separable? ;) – Yemon Choi Feb 21 at 6:16

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Yemon's answer shows that every separable Frechet space is compactly generated. The converse is also true: The compact generator is contained in the closed absolutely convex hull of a sequence converging to $0$, and the countable set of all rational (finite) linear combinations of that sequence is dense. – Jochen Wengenroth Feb 21 at 10:18

Thanks for the quick explanation! – Iian Smythe Feb 21 at 15:22

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@Yemon: I believe that space is not bad, it is misunderstood. – Nik Weaver Feb 21 at 18:32

Compactly generated Banach spaces

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