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Question on dense subset of l^p space

  1. Jul 16, 2015 #1
    Let [itex]X[/itex] be an infinite set. Consider the set [itex]l^p(X)[/itex], where [itex]1\leq p < +\infty[/itex], of all complex functions that satisfy the inequality
    [tex]\sup \{\sum_{x\in E} |f(x)|^p: E \subset X, \;\; |E|<\aleph_0 \} < +\infty [/tex].
    The function [itex]\| \|_p: l^p(X)\rightarrow \mathbb{[0,+\infty]}[/itex] defined by
    [tex]\| f \|_p = \sup \{ \left( \sum_{x\in E} |f(x)|^p \right)^{1/p}: E \subset X, \; |E|<\aleph_0 \}[/tex]
    makes [itex]l^p(X)[/itex] a complete normed vector space.

    What I'm trying to show is that there exists a dense subset of [itex]l^p(X)[/itex] with cardinality equal to that of [itex]X[/itex]. For every point [itex]a\in X[/itex] we consider the function [itex]\delta_a: X\rightarrow \mathbb{C}[/itex] with [itex]\delta_a(x) = 0[/itex], if [itex]x\neq 0[/itex], and [itex]\delta_a (a) = 0[/itex].

    Let [itex]f\in l^p(X)[/itex]. Consider the collection [itex]\mathcal{C}[/itex] of all finite subsets of [itex]X[/itex]. The relation [itex]\subset[/itex] on [itex]\mathcal{C}[/itex] makes this collection a directed set. For every [itex]E\in\mathcal{C}[/itex], let [itex]g_E = \sum_{a\in E} f(a) \delta_a[/itex]. The mapping [itex]E\rightarrow g_E[/itex] constitutes a net, which I'm trying to show that it converges to [itex]f[/itex].

    If [itex]\epsilon>0[/itex] there exists a finite subset [itex]E[/itex] of [itex]X[/itex] such that
    [tex]\|f\|_p - \left(\sum_{x\in E}|f(x)|^p \right)^{1/p}<\epsilon[/tex]
    which in turn leads to
    [tex]\| f \|_p - \| g_E \|_p <\epsilon.[/tex]Therefore, if [itex]G[/itex] a finite subset of [itex]X[/itex] that contains [itex]E[/itex] then [itex]\| f \|_p - \| g_G \|_p < \epsilon [/itex]. What I've been having problem proving is the inequality [itex] \| f - g_G \|_p < \epsilon [/itex]. Any ideas on this? Thanks in advance! :)
     
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  3. Jul 16, 2015 #2

    micromass

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    How would you prove it for ##X = \mathbb{N}##? Can you mimic that proof?
     
  4. Jul 17, 2015 #3

    mathman

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    There seems to be at least one serious typo. [itex]\delta_a(a)=0[/itex] Should be =1?
     
  5. Jul 18, 2015 #4

    WWGD

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    Together with unstated conditions:
    1)What is ##E##
    2) I think the condition ## || ||_p ## should have range in ## [0, \infty) ##. I am not aware of infinity-valued normed spaces.
    3) This may be obvious, but I guess your bet is defined on $l^p(X)$, and not on, some other set?
     
  6. Jul 18, 2015 #5

    micromass

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    ##E## is finite subset of ##X##.

    It does since ##E## is finite.
     
  7. Jul 18, 2015 #6

    WWGD

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    Ah, yes, my bad, I missed the "## |E| < \aleph_0 ##"; I expected a sort of ## |E| < \infty ##; clearly finite sums are bounded/convergent.
     
  8. Jul 20, 2015 #7
    Hi https://www.physicsforums.com/threads/question-on-dense-subset-of-l-p-space.823629/members/kostas230.419693/ [Broken],

    probably the easiest way is to show that finite functions (i.e. functions that are non-zero at finitely points) taking only rational values are dense.
    Can you prove that?
     
    Last edited by a moderator: May 7, 2017
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