Saturday, 11 June 2016

proof writing - Trouble in proving that |x|p=max|xj|




We define p-norm in this way: xp={Nj=1|xj|p}1p
We know that It change to xp=max when p \to \infty
How can I prove this ?


Answer




What we have to prove is that if x_1,\dots,x_K are real numbers such that 0\leqslant x_k\lt 1, then
\lim_{p\to \infty}\left(1+\sum_{k=1}^Kx_k^p\right)^{1/p}=1.
It's equivalent to show that
\lim_{p\to \infty}\frac 1p\log\left(1+\sum_{k=1}^Kx_k^p\right)=0.
It's the case, since for t\geqslant 0, 0\leqslant\log(1+t)\leqslant t.


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