State the Lagrangian sufficiency theorem.

Let $p \in(1, \infty)$ and let $a_{1}, \ldots, a_{n} \in \mathbb{R}$. Maximize

$$\sum_{i=1}^{n} a_{i} x_{i}$$

subject to

$$\sum_{i=1}^{n}\left|x_{i}\right|^{p} \leqslant 1, \quad x_{1}, \ldots, x_{n} \in \mathbb{R}$$