1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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).
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook