Minkowski's Inequality for Integrals

Theorem
Let $f, g$ be integrable functions in $X \subseteq \R^n$ with respect to the volume element $dV$.


 * $(1):\quad$ Let $p > 1$. Then:
 * $\displaystyle \left({\int_X \left\vert{f + g}\right\vert^p \mathrm d V}\right)^{1/p} \le \left({\int_X \left\vert{f}\right\vert^p \mathrm d V}\right)^{1/p} + \left({\int_X \left\vert{g}\right\vert^p \mathrm d V}\right)^{1/p}$
 * $(2):\quad$ Let $p < 1, p \ne 0$. Then:
 * $\displaystyle \left({\int_X \left\vert{f + g}\right\vert^p \mathrm d V}\right)^{1/p} \ge \left({\int_X \left\vert{f}\right\vert^p \mathrm d V}\right)^{1/p} + \left({\int_X \left\vert{g}\right\vert^p \mathrm d V}\right)^{1/p}$