the problem I'm stuck on is the following:
Suppose that S is a countably infinite subset of $\ell_2$ with the property that the linear span of S′ is dense in $\ell_2$ whenever S\S′ is finite. Show that there is some S′ whose linear span is dense in $\ell_2$ and for which S\S′ is infinite.
I have tried repeatedly to solve this in somewhat of a 'bang my head against a wall' manner, by constructing a series of subsets of some arbitrary S, such that the complement is finite and of increasing size, but I haven't had any success. I haven't actually used the fact that we're working in $\ell_2$ here, so it's quite likely that I'm meant to use some property of Hilbert spaces - however, I'm not sure what. Could anyone please help?
Thankyou very much; Stephen.