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Thread: triangle inequality

  1. #1
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    triangle inequality

    Hey guys.
    How can I prove the triangle inequality for this norm.
    I marked in green what I did but then I got stuck.

    Thanks in advance.
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  2. #2
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    This is Minkowski's inequality. You can find proofs online if you Google for them.

    The basic idea is start with
    \begin{aligned}\|x+y\|_p^p = \sum|x_i+y_i|^p &\leqslant \sum\bigl(|x_i|+|y_i|\bigr)|x_i+y_i|^{p-1} \\ &= \sum\bigl|x_i||x_i+y_i|^{p-1} + \sum|y_i||x_i+y_i|^{p-1},\end{aligned}
    and then use Hölder's inequality on each of those two sums.
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  3. #3
    Moo
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    Hello,
    Quote Originally Posted by asi123 View Post
    Hey guys.
    How can I prove the triangle inequality for this norm.
    I marked in green what I did but then I got stuck.

    Thanks in advance.
    Consider \|x+y\|_p^p=\sum_{i=1}^\infty |x_i+y_i|^p
    And then you know that |m+n| \leqslant |m|+|n|
    So \forall i \geqslant 1,~ |x_i+y_i|\leqslant |x_i|+|y_i|, which implies |x_i+y_i|^p \leqslant (|x_i|+|y_i|)^p

    And hence \|x+y\|_p^p=\sum_{i=1}^\infty |x_i+y_i|^p \leqslant \sum_{i=1}^\infty (|x_i|+|y_i|)^p

    Now take the p-th root and you're done.
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  4. #4
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    Quote Originally Posted by Moo View Post
    take the p-th root and you're done.
    Do what?

    Thanks a lot.
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  5. #5
    Moo
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    Quote Originally Posted by asi123 View Post
    Do what?

    Thanks a lot.
    Nah >< I'm really sorry, I thought you wanted to prove the green stuff

    (p-th root of x is x^{1/p})
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