Consider the measure space \((\Omega=\{1,2, \ldots, n\}, \mathscr{P}(\Omega), \mu), n \geqslant 2\), where \(\mu\) is the counting measure. Show that \(\left(\sum_{i=1}^{n}\left|x_{i}ight|^{p}ight)^{1 / p}\) is a norm if \(p \in[1, \infty)\), but not for \(p \in(0,1)\). [ you can identify \(\mathcal{L}^{p}(\mu)\) with \(\mathbb{R}^{n}\).]