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Attached is a proof I found. It is probably very basic, but I can not understand the marked thing. Why this term is zero? I hope someone can explain it for me. enter image description here

Edit(elaboration): A Norm is a function that takes a function $f$ and returns a number. Discreet norm's input is not the function itself but it's values at certain defined points. Each discreet norm has it's own set of points $\{x_i\}$ (and also weights $\{w_i\}$ ).

There are some conditions it should follow to be called a norm (you can google it).

Here, $f_i$ is short notation of $f(x_i)$, and the $L_p$ norm is defined as $L_p \equiv (\sum{|f_i|^p w_i})^{1/p} $

$|f_i|$ is simply absolute value. And of course $f$ should be defined at $\{x_i\}$ points.

If we send $p$ to infinity then we get the infinity norm $L_\infty$.

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    @WNY Thank you. The squeeze theorem makes more sense.2012-02-15

2 Answers 2

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Let $x\in R^n$ and $\|x\|_\infty=\max_{1\leq i\leq n}|x_i|$, write $\|x\|_p$ as $ \|x\|_p = \left(\sum_{i=1}^n|x_i|^p\right)^{1/p}=\|x\|_\infty\left(\sum_{i=1}^n\left(\frac{|x_i|}{\|x\|_\infty}\right)^p\right)^{1/p} $

noting that $\left(\frac{|x_i|}{\|x\|_\infty}\right)^p\leq1$ for every $i$, with equality at least once and at most $n$ times, then $ \|x\|_\infty\leq\|x\|_p\leq \|x\|_\infty n^{1/p} $ and because $n>0$ gives $\lim_{p\to\infty}n^{1/p}=1$, then $\lim_{p\to\infty}\|x\|_\infty$.

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    @brdcastguy Imagine we could re-write the p-norm of $x$ in terms of whichever component happens to be the $||x||_{\infty}$. This would say $||x||_{p} = (||x||_{\infty}^{p} + |x_2|^{p} + ...)^{1/p}$. Since the p-th root function is strictly increasing along the positive real axis, and we're only considering absolute values of input components, this means it is greater than $(||x||_{\infty}^{p})^{1/p} = ||x||_{\infty}$. This is for positive powers $p$, since we care about $p$ defining an $L_{p}$ norm.2018-04-09
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$(f_i/M)^p$ tends to zero becouse of exponent p tends to +infinite and $f_i/M<1$.

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    This is part of one half of the proof.2012-10-17