Is l2 Space Separable and Second Countable?

[complexnumber]

Homework Statement

1. Prove that if a metric space $$(X,d)$$ is separable, then
$$(X,d)$$ is second countable.2. Prove that $$\ell^2$$ is separable.

Homework Equations

The Attempt at a Solution

1. $$\{ x_1,\ldots,x_k,\ldots \}$$ is countable dense subset. Index the
basis with rational numbers, $$\{ B(x,r) | x \in A, r \in \mathbb{Q} \}$$ is countable (countable $$\times$$ countable).

2. What set is a countable dense subset of $$\ell^2$$?

 

[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 $$\ell^2$$ by sequences in A, hence the closure is $$\ell^2$$. (The set $$\ell^2$$ \ 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 $$\frac{\epsilon}{N}$$ of it.
Then the distance is $$N*\frac{\epsilon}{N}$$.

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).