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Hölder's inequality for sequences.

  1. Dec 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Let [itex]1\leq p,q[/itex] that satisfy [itex]p+q=pq[/itex] and [itex]x\in\ell_{p},\, y\in\ell_{q}[/itex]. Then
    [itex]
    \begin{align}
    \sum_{k=1}^{\infty}\left\vert x_{k}y_{k}\right\vert\leq\left(\sum_{k=1}^{\infty}\left\vert x_{k}\right\vert^{p}\right)^{\frac{1}{p}}\left( \sum_{k=1}^{\infty}\left\vert y_{k}\right\vert^{q}\right)^{\frac{1}{q}}
    \end{align}
    [/itex]
    2. Relevant equations

    The Hölder's inequality for [itex]\mathbb{R}^{n}[/itex] and convergence conditions of sequences in [itex]\ell_{r}[/itex], that is:
    [itex]
    \begin{align}
    \sum_{k=1}^{\infty}\left\vert x_{k}\right\vert^{r}<\infty
    \end{align}
    [/itex]

    3. The attempt at a solution

    I can prove the result from the inequality for [itex]\mathbb{R}^{n}[/itex], but I have a missing part that I don't get to prove, that is: proving that
    [itex]
    \begin{align}
    \sum_{k=1}^{\infty}\left\vert x_{k}y_{k}\right\vert
    \end{align}
    [/itex]
    converges given convergence conditions over x, y. Could you give me ideas! This is not a homework task. I'm reviewing some analysis topics.
     
  2. jcsd
  3. Dec 28, 2011 #2

    micromass

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    How did you prove the inequality for [itex]\mathbb{R}^n[/itex]?? Can you adapt the proof?
     
  4. Dec 28, 2011 #3
    The problem I have is not the proof itself, but the convergence of the LHS of the inequality. How can I prove it is the question. Obviously in [itex]\mathbb{R}^{n}[/itex] you don't need to check any convergence, then you have no manner to parallel that part of the proof.

    In other words: I have followed the proof for [itex]\mathbb{R}^{n}[/itex] and proven the inequality for sequences, but I failed to justify why I can do it since I don't know how to prove that if the series for x, y converge with the convergence condition for that sequence spaces then the series in LHS converges.
     
  5. Dec 28, 2011 #4
    The result for [itex]\mathbb{R}^{n}[/itex] is

    Let [itex]1\leq p,q[/itex] that satisfy [itex]p+q=pq[/itex] and [itex]x,y\in\mathbb{R}^{n}[/itex]. Then
    [itex]
    \begin{align}
    \sum_{k=1}^{n}\left\vert x_{k}y_{k}\right\vert\leq\left(\sum_{k=1}^{n}\left\vert x_{k}\right\vert^{p}\right)^{\frac{1}{p}}\left( \sum_{k=1}^{n}\left\vert y_{k}\right\vert^{q}\right)^{\frac{1}{q}}
    \end{align}
    [/itex]
     
  6. Dec 28, 2011 #5

    micromass

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    Maybe you can start by proving that

    [tex]\frac{|x_ny_n|}{\|(x_n)_n\|_p\|(y_n)_n\|_q}\leq \frac{1}{p}\frac{|x_n|^p}{\|(x_n)_n\|_p}+\frac{1}{q}\frac{|y_n|^q}{\|(y_n)_n\|_q}[/tex]

    In general if [itex]0<\lambda <1[/itex] and a,b are nonnegative, then

    [tex]a^\lambda b^{1-\lambda}\leq \lambda a+(1-\lambda)b[/tex]
     
  7. Dec 28, 2011 #6
    Why I can't see that!! that's another version of the Young's inequality. thanks for that illuminating idea.
     
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