1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    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!

Convergence in L{p} space

  1. Nov 19, 2013 #1
    If [itex]f_{n} \underset{n \to \infty}{\longrightarrow} f[/itex] in [itex]L^{p}[/itex], [itex]1 \leq p < \infty[/itex], [itex]g_{n} \underset{n \to \infty}{\longrightarrow} g[/itex] pointwise and [itex]|| g_{m} ||_{\infty} \leq M \forall n \in \mathbb{N}[/itex] prove that:
    [itex]f_{n} g_{n} \underset{n \to \infty}{\longrightarrow} fg[/itex] in [itex]L^{p}[/itex]

    My attemp:

    [itex]\displaystyle \int | f_{n} g_{n} - f g_{n} |^{p} = \displaystyle \int | g_{n} | | f_{n} - f |^{p} \leq M^{p} \displaystyle \int | f_{n} - f |^{p}[/itex]
    Then [itex]f_{n} g_{n} \underset{n \to \infty}{\longrightarrow} f g_{n}[/itex] in [itex]L^{p}[/itex]
    Now lets prove that [itex]f g_{n} \underset{n \to \infty}{\longrightarrow} fg [/itex] in [itex]L^{p}[/itex]
    [itex]g_{n} \longrightarrow g[/itex] a.e. [itex]\Longrightarrow g_{n} \underset{\longrightarrow}{m} g[/itex]
    [itex]\forall \varepsilon > 0, \forall \delta > 0, \exists n_{0} \in \mathbb{N} / \forall n \geq n_{0} :[/itex]
    [itex]| D | = | \{ x / | g_{n} (x) - g(x) | \geq \delta \} | < \varepsilon[/itex]

    [itex]\displaystyle \int | f g_{n} - fg |^{p} = \displaystyle \int | f |^{p} | g_{n} - g |^{p} \leq \displaystyle \int_{D} | f |^{p} M^{p} + \displaystyle \int_{D^{c}} | f |^{p} \delta^{p}[/itex]
    I know [itex]| D | < \varepsilon[/itex], but [itex]f[/itex] isn't necessarily bounded in [itex] D [/itex], I need to prove that [itex]\int_{D} | f |^{p} \longrightarrow 0[/itex] as [itex]| D | \to 0[/itex]

    Anyone has any idea?
    Is this approach right or the last step is false and I need to rework the proof enterely?

    Edit 2: Edit 1 was completely wrong so I deleted it.
     
    Last edited: Nov 19, 2013
  2. jcsd
  3. Nov 19, 2013 #2
    You'll need to redo the proof start by adding an additional term and immediately subtracting it then use your bounds and the fact that gn => g and fn=>f pointwise.
     
  4. Nov 19, 2013 #3
    You mean to be able to apply dominated convergence theorem?

    Anyway, I asked my teacher and he knew filled the last gap in my proof.

    As
    [itex]\displaystyle \int | f |^{p} < \infty \Longrightarrow \displaystyle \int_{E} | f |^{p} < \infty \forall E \subset R^{n}[/itex]
    Then by the absolute continuity of the Lebesgue integral, for every [itex]\varepsilon > 0[/itex] there exists a [itex]\delta ' > 0 [/itex] such that:
    [itex]| E | < \delta 0 \Longrightarrow | \displaystyle \int_{E} | f |^{p} | < \varepsilon [/itex]
    As [itex](2M)^{p}[/itex]* is a constant, this converges to zero as [itex]\delta \to 0[/itex] and [itex]\varepsilon \to 0[/itex]
    * I made a little mistake in the previous post:
    [itex]| g - g_{n} |^{p} \leq (2M)^{p}[/itex] not only [itex]M^{p}[/itex]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Convergence in L{p} space
  1. [tex]l^{p} space[/tex] (Replies: 5)

Loading...