The $L^p$ norm (or $\ell_p$ norm) is a mathematical function that measures the size or length of a vector. For a vector $\mathbf{x} \in \mathbb{R}^n$ and a real number $p \geq 1$, the $L^p$ norm is defined as: $\left\|\mathbf{x}\right\|_p = \left( \sum_{i=1}^n |x_i|^p \right)^{1/p}$ Common vector norms are special cases of the $L^p$ norm: - The $L_1$ norm is obtained by setting $p=1$. - The $L_2$ norm (Euclidean norm) is obtained by setting $p=2$. - The max norm ($L^\infty$ norm) is obtained in the limit as $p \to \infty$, resulting in: $\left\|\mathbf{x}\right\|_\infty = \max_{i} |x_i|$