I am missing something in the identity $(1.18)$ below. What does that identity have to do with the fact that the $L^p$ norm is a non-negative number? The identity $(1.18)$ says that when we raise $f$ to some positive power $r$, we raise the norm to the corresponding power $r$, but now the norm is no longer the $L^p$ norm, but rather the $L^{pr}$ norm. Could someone explain the relation between those two sentences, or give an intuitive explanation of what is going on?
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I guess the point is that this wouldn't hold if the norm weren't a non-negative number. Indeed if, say, $\| f^2 \|_{L^p}$ were negative, then clearly $\| f^2 \|_{L^p} \neq \| f \|_{L^{2p}}^2$. The argument fails for the same reason the following reasoning fails: $$-1 = (-1)^1 = ((-1)^2)^{1/2} = 1.$$
Just write out the norms: $$\|f^r\|_{L^p}=\left(\int_X|f^r|^p d\mu\right)^{1/p}=\left(\int_X|f|^{pr} d\mu\right)^{1/p}=\left(\left(\int_X|f|^{pr} d\mu\right)^{1/pr}\right)^r=\|f\|_{L^{pr}}^r$$ So in particular if $f\in L^p$ then $f^r\in L^{p/r}$.