Is the mapping from $\mathcal{L}^p(X,\mu)$ to $\mathcal{L}^1(X,\mu)$ continuous?

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Here is this week's POTW:

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Let $(X,\mu)$ be a positive measure space. For $0 < p < \infty$, why is the mapping $\mathcal{L}^p(X,\mu) \to \mathcal{L}^1(X,\mu)$ sending $f$ to $\lvert f\rvert^p$, continuous?-----Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

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No one answered this week's problem. You can read my solution below.

Spoiler

For all pairs of nonnegative numbers $a, b$, $\lvert a^p - b^p\rvert \le \lvert a - b\rvert^p$ if $0 < p < 1$, and $\lvert a^p - b^p\rvert \le p(a^{p-1} + b^{p-1})\lvert a - b\rvert$ if $p \ge 1$. Thus, for all $f, g\in \mathcal{L}^p(X,\mu)$,

$$\| \lvert f\rvert^p - \lvert g\rvert^p \|_1 \le \|f - g\|_p^p \quad (0 < p < \infty)$$

and for $1\le p < \infty$,

$$\|\lvert f\rvert^p - \lvert g\rvert^p\|_1 \le p\int_X \lvert f\rvert^{p-1}\lvert f - g\rvert\, d\mu + p\int_X \lvert g\rvert^{p-1}\lvert f - g\vert\, d\mu \le p(\|f\|_p\|f - g\|_p + \|g\|_p\|f - g\|_p) = p(\|f\|_p + \|g\|_p)\|f - g\|_p$$

using Hölder's inequality in the penultimate step. These inequalities imply continuity of the map $f\mapsto \lvert f\rvert^p$ from $\mathcal{L}^p(X,\mu)$ to $\mathcal{L}^1(X,\mu)$.