Closure of $ell^2$ in the space of real sequences Code Answer

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Using the product topology on $overline{mathbb{R}}^omega$, is $ell^2$ (the space of real square summable sequences) a dense subset of $overline{mathbb{R}}^omega$ ?

Answer

$ell_2$ contains all sequences with only finitely many non-zero components: $$D = {(x_n) in mathbb{R}^omega: |{n: x_n neq 0}| < aleph_0 }subset ell_2$$

$D$ is clearly dense in $mathbb{R}^omega$, so $ell_2$ is as well.

The same holds for $overline{mathbb{R}}^omega$ in the product topology. (This contains $mathbb{R}^omega$ as a dense subset and “dense in dense is dense”.

So the closures are the whole space in both cases.

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