Lemma 15.2.2.label Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, $p \in [1, \infty)$, and $f, g \in L^{p}(X; E)$, then
\[|\norm{f}_{L^p(X; E)}^{p} - \norm{g}_{L^p(X; E)}^{p}| \le p\norm{f - g}_{L^p(X; E)}(\norm{f}_{L^p(X; E)}\vee \norm{g}_{L^p(X; E)})^{p-1}\]
Proof. By Lemma 3.2.2,
\begin{align*}|\norm{f}_{L^p(X; E)}^{p} - \norm{g}_{L^p(X; E)}^{p}|&\le p|\norm{f}_{L^p(X; E)}- \norm{g}_{L^p(X; E)}|\\&\times (\norm{f}_{L^p(X; E)}\vee \norm{g}_{L^p(X; E)})^{p-1}\\&\le p\norm{f - g}_{L^p(X; E)}(\norm{f}_{L^p(X; E)}\vee \norm{g}_{L^p(X; E)})^{p-1}\end{align*}
$\square$
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