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Let $x \in \mathbb{R}^n$ and $$\|x\|_p= \left(\sum_{i=1}^n |x_i|^p\right)^ {1/p}\ $$ for $0< p < \infty $ and $\|x\|_\infty=\max_{i=1,...,n}|x_i|$. Show that $\lim_{p\to 0} \|x\|_p$ exists and determine its value (we also allow infinity as a limit).

To begin with, I don't fully grasp the intuition of this problem. Why is that they want to determine this limit for zero?

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    $\begingroup$ in analysis, in general, there is no intuition. This is why we need to do analysis, because intuition fails so often for very abstract settings. Also this is the same reason why mathematics developed slowly the last 3000 years. $\endgroup$
    – Masacroso
    Apr 18, 2019 at 6:09

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When $n=1$, obviously $||x||_p = |x_1|$ which is the limit in question. Now assume that $n>1$ and $x = (x_1,...,x_n)\in \mathbb{R}^n$ has at least one non-zero coordinate (otherwise the limit is $0$).

In case $x$ has only one non-zero coordinate, the problem reduces to the case of $n=1$, and the limit becomes equal to the absolute value of that non-zero coordinate. Hence we will assume that $x$ has at least two non-zero coordinates, and will prove that in this case the limit is infinite. Indeed, assume without loss of generality that $|x_1| \geq |x_i|$ for all $i=2,...,n$ and that $x_2 \neq 0$. Then $$ ||x||_p = |x_1| \left( 1 + \frac{|x_2|^p}{|x_1|^p} +...+ \frac{|x_n|^p}{|x_1|^p} \right)^{1/p} \geq |x_1| \left( 1 + \frac{|x_2|^p}{|x_1|^p} \right)^{1/p} := |x_1| \left( 1 + t^p \right)^{1/p}, \tag{1} $$ where $t = |x_2|/|x_1| $ and by our assumption $0<t \leq 1$. We claim that $(1 + t^p)^{1/p} \to \infty$ as $p \to 0$, which clearly implies by $(1)$ that $||x||_p \to \infty$ as $p\to 0$. To prove the claim we use the inequality $$ \log(1 + t ) \geq t - \frac{t^2}{2}, \text{ for all } 0 \leq t \leq 1. \tag{2} $$ To see $(2)$, consider the function $f(t) = \log(1 + t) - t + t^2/2$ in $[0,1]$ and differentiate twice to see that it increases.

Now, using $(2)$ we get $$ \log( 1 + t^p)^{1/p} \geq \frac{1}{p} \left( t^p - \frac{t^{2p}}{2} \right) = \frac{t^p}{p} \left( 1 - \frac{t^p}{2} \right) \to \infty \text{ as } p\to 0. $$ This completes the proof that $||x||_p \to \infty$ when $p\to 0$, as long as $x$ has at least two non-zero coordinates.

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