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Homework Help: Show that l2 space separable

  1. May 12, 2010 #1
    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]?
     
  2. jcsd
  3. May 17, 2010 #2
    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.
     
  4. May 17, 2010 #3

    Landau

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    Science Advisor

    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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