We define p-norm in this way: ‖x‖p={∑Nj=1|xj|p}1p
We know that It change to ‖x‖p=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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