text: # Show that l2 space separable
  
  1. May 12, 2010
  
  ### complexnumber
  
  1. The problem statement, all variables and given/known data
  
  1. Prove that if a metric space $$(X,d)$$ is separable, then
  $$(X,d)$$ is second countable.
  
  2. Prove that $$\ell^2$$ is separable.
  
  2. Relevant equations
  
  3. 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 $$\ell^2$$?
  
  2. May 17, 2010
  
  ### 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.
  
  3. May 17, 2010
  
  ### 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).
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