Constructing a subset of l_2 with dense linear span with finite complement

[Mathmos6]

Homework Statement

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.

The Attempt at a Solution

I have tried repeatedly to solve this by constructing a series of subsets of some arbitrary S, such that the complement is finite and of increasing size. 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; Mathmos6

 

[mighty2000]

Dear Mathmos6,

Thank you for your post. This is a very interesting problem and I appreciate your attempt at solving it. Let me provide some guidance on how to approach this problem.

First, let's recall the definition of a dense subset. A subset S of a topological space X is said to be dense if every point in X is either in S or a limit point of S. In other words, every open set in X contains a point from S.

Now, let's consider the linear span of a set S in \ell_2. This is defined as the set of all finite linear combinations of elements in S. In other words, it is the set of all possible sums of the form a_1x_1 + a_2x_2 + ... + a_nx_n, where a_i are scalars and x_i are elements in S.

Now, let's think about the given property of S. We know that the linear span of S′ is dense in \ell_2 whenever S\S′ is finite. This means that for any point x in \ell_2, we can find a sequence of points in the linear span of S′ that converges to x. In other words, the linear span of S′ is dense in \ell_2.

With this in mind, let's try to construct a set S′ that satisfies the given property. We can start with any countably infinite subset S of \ell_2. Now, we want to find a subset S′ of S such that the linear span of S′ is dense in \ell_2 and S\S′ is infinite.

One way to do this is to consider the complement of S in \ell_2. This complement is an uncountable set, so we cannot simply choose a finite subset from it. However, we can use the fact that \ell_2 is a separable space. This means that there exists a countable dense subset D of \ell_2. In other words, every point in \ell_2 is either in D or a limit point of D.

Now, let's consider the complement of S in \ell_2. This complement is an uncountable set, but we know that it contains a countable dense subset D. We can choose a countably infinite subset S′' of D and add it to S to form a new set S′ = S ∪ S′