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Math Help - Lp-norm converges to Chebyshev norm?

  1. #1
    Junior Member
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    Lp-norm converges to Chebyshev norm?

    My textbook says that the Lp-norm converges to the Chebyshev norm. This seems intuitive, but I fail to prove it.

    Thus, my question is, how do I prove that \lim_{p\rightarrow \infty} (|x_1|^p+|x_2|^p+\dots+|x_n|^p)^{1/p} = \max(|x_1|,\dots,|x_n|)
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  2. #2
    Senior Member roninpro's Avatar
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    Without loss of generality, take |x_1| to be the maximum of all of the |x_i|. Try working with

    (|x_1|^p+|x_2|^p+\dots+|x_n|^p)^{1/p}=|x_1|\left(1+\left(\frac{|x_2|}{|x_1|}\right)^p  +\left(\frac{|x_3|}{|x_1|}\right)^p+\ldots+\left(\  frac{|x_n|}{|x_1|}\right)^p\right)^{1/p}

    noting that \frac{|x_i|}{|x_1|}\leq 1.
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