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Problem on seperable banach spaces

  1. Jul 19, 2009 #1
    Prove that if a Banach space X, has separable dual X*, then X is separable.

    It gives the hint that the first line of the proof should be to take a countable dense subset [tex]\{f_n\}[/tex] of X* and choose [tex]x_n\in X[/tex] such that for each n, we have [tex]||x_n||=1[/tex] and [tex]|f_n(x)|\geq(1/2)||f_n||[/tex].

    Ok so what do I do now. We want to show that X is separable, so it's countable dense subset would be [tex]\{x_n\}[/tex], which we just have to show is dense in X, how do I do this?
     
  2. jcsd
  3. Jul 19, 2009 #2
    Those [tex]x_n[/tex] won't be dense (they all have norm 1). Instead show their linear span is dense, then argue that [tex]X[/tex] is separable.
     
  4. Jul 19, 2009 #3

    HallsofIvy

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    Given any x in X, there exist a corresponding x* in X*. Since {x*} is dense in X*, for any [itex]\epsilon> 0[/itex], there exist a x* such that ||X*- x*||< [itex]\epsilon[/itex]. Let x* be the member of X corresponding to x*.
     
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