Lp-norm converges to Chebyshev norm?

**batman** · November 12th 2010, 12:52 AM

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 $\lim_{p\rightarrow \infty} (|x_1|^p+|x_2|^p+\dots+|x_n|^p)^{1/p} = \max(|x_1|,\dots,|x_n|)$

**roninpro** · November 12th 2010, 08:53 AM

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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