Definition:Hölder Mean/Positive Infinite Exponent

Definition
Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be positive real numbers.

Let $p$ be an extended real number.

Let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$.

For $p = \infty$, the Hölder mean is defined as:
 * $\map {M_\infty} {x_1, x_2, \ldots, x_n} = \max \set {x_1, x_2, \ldots, x_n}$

Also see

 * Limit of Hölder Mean as Exponent tends to Infinity, which justifies this definition