Definition:Max Operation/General Definition/Real Numbers

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Definition

The max operation on the real cartesian space $\R^n$ is the real-valued function $\max: \R^n \to \R$ defined recursively as:

$\forall x := \left \langle{x_i}\right \rangle_{1 \mathop \le i \mathop \le n} \in \R^n: \max \left({x}\right) = \begin{cases} x_1 & : n = 1 \\ \max \left({x_1, x_2}\right) & : n = 2 \\ \max \left({\max \left({x_1, \ldots, x_{n-1}}\right), x_n}\right) & : n > 2 \\ \end{cases}$

where $\max \left({x, y}\right)$ is the binary max operation on $\R \times \R$.