(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

1. Prove that if a metric space [tex](X,d)[/tex] is separable, then

[tex](X,d)[/tex] is second countable.

2. Prove that [tex]\ell^2[/tex] is separable.

2. Relevant equations

3. The attempt at a solution

1. [tex]\{ x_1,\ldots,x_k,\ldots \}[/tex] is countable dense subset. Index the

basis with rational numbers, [tex]\{ B(x,r) | x \in A, r \in \mathbb{Q}

\}[/tex] is countable (countable [tex]\times[/tex] countable).

2. What set is a countable dense subset of [tex]\ell^2[/tex]?

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# Homework Help: Show that l2 space separable

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