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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.

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    I was stuck on this problem as well. Try using Gram-Schmidt. (The intuition here comes from the rank-nullity theorem: in finite-dimensional vector spaces, the cardinality of the largest set of vectors you can remove from a given set S of vectors and still have it span the entire space is related to the rank of a matrix you can build out of S. For Hilbert spaces there is a good notion of rank, and it comes from finding orthonormal bases.)2011-01-17
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    @Qiaochu Yuan: So $S = \{x_i: \sum x_{i}^{2} \ \text{converges for} \ i = 1, \dots \}$?2011-01-17
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    @Trevor: I don't understand your comment. $\ell_2$ is the set of all sequences $(x_1,x_2,\ldots)$ such that $\sum_i|x_i|^2\lt\infty$, so in particular $\sum_ix_i^2$ always converges.2011-01-17
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    It should be noted that though the title of the question is very clear, the question is not. The quantity $S'$ is introduced without defining it, and then it is asked to find it. Does the question really reads as 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 $\ell_2\backslash S$ is finite. Show that there is some $S'$ whose linear span is dense in $\ell_2$ and for which $\ell_2\backslash S'$ is infinite?2011-01-17
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    No, that is not the question. The question is about a set S with dense linear span such that removing any finite subset of S leaves a set that still has dense linear span. You are supposed to show that then necessarily S has some infinite subset S' such that S\S' still has dense span.2011-01-17

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