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Math Help - How to prove Hölders inequality for sums?

  1. #1
    Ase
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    How to prove Hölders inequality for sums?

    Hi

    I am trying to prove Hölders inequality for sums but I can't get any further. So far I have proven Young's inequality and then I got stuck. Can anyone help?

    Thanks in advance.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Ase View Post
    Hi

    I am trying to prove Hölders inequality for sums but I can't get any further. So far I have proven Young's inequality and then I got stuck. Can anyone help?

    Thanks in advance.
    You need to be more specific. I assume you're talking about \sum_{k=1}^{n}|x_ky_k|\leqslant\sqrt[\frac{1}{p}]{\sum_{k=1}^{n}|x_k|}\sqrt[\frac{1}{q}]{\sum_{k=1}^{n}|y_k|} with \frac{1}{p}+\frac{1}{q}=1. There is the alternative one for infinite series. Which are you after?
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  3. #3
    Moo
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    Take the measure theoric version of Hölder's inequality and apply it with a counting measure
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  4. #4
    Ase
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    Hi Drexel28

    I'm sorry that I did'nt post a specifik formular (I am completely new to this forum and did'nt know how.) I am refering to the one for infinite series.

    Thanks in advance

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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Ase View Post
    Hi Drexel28

    I'm sorry that I did'nt post a specifik formular (I am completely new to this forum and did'nt know how.) I am refering to the one for infinite series.

    Thanks in advance

    Well, have you proved the finite case?

    We know that for all fixed finite m\in\mathbb{N} that \sum_{n=1}^{m}|x_ny_n|\leqslant\left(\sum_{n=1}^{m  }|x_n|\right)^{\frac{1}{p}}\left(\sum_{n=1}^{m}|y_  n|\right)^{\frac{1}{q}}. Now apply the squeeze theorem and remember that we define \sum_{n=1}^{\infty}a_n to be the limit of the partial sums.
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  6. #6
    Ase
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    Thanks a lot Drexel28. I now have some clues on how to solve the problem (and no I did'nt prove it for finite sums).
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