Is l2 Space Separable and Second Countable?

[complexnumber]

Homework Statement

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.

Homework Equations

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]?

 

[ninty]

2. Let A = the set of sequences with only finitely many non-zero components(N of them), where each term is a member of the rationals.
We can show that the we can approximate every element of [tex] \ell^2 [/tex] by sequences in A, hence the closure is [tex] \ell^2 [/tex]. (The set [tex] \ell^2 [/tex] \ A are the limit points)
If you think about it, between any reals there's a rational number
So for each term, we can get a rational that is of distance [tex] \frac{\epsilon}{N} [/tex] of it.
Then the distance is [tex]N*\frac{\epsilon}{N}[/tex].

Take limit as N goes to infinity.

It's late here so I'm not really capable of putting all this into nice sentences.

 

[Landau]

1. correct
2. this comes down to the fact that R (or C) is separable; just restrict to rationals and finite sequences (see ninty's reply).