Definition:Pointwise Supremum of Extended Real-Valued Functions

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Definition

Let $S$ be a set, and let $\left({f_i}\right)_{i \in I}, f_i: S \to \overline \R$ be an $I$-indexed collection of extended real-valued functions.


Then the pointwise supremum of $\left({f_i}\right)_{i \in I}$, denoted $\displaystyle \sup_{i \in I} f_i: S \to \overline{\R}$, is defined by:

$\displaystyle \left({\sup_{i \in I} f_i}\right) \left({s}\right) := \sup_{i \in I} f_i \left({s}\right)$

where the latter supremum is taken in the extended real numbers $\overline \R$.


By Extended Real Numbers form Complete Poset, this supremum is guaranteed to exist.


Thence it can be seen that pointwise supremum is an instance of a pointwise operation on extended real-valued functions.


Also known as

Because of the way $\displaystyle \sup_{i \in I} f_i$ is defined, there is usually no need to distinguish between the left- and right-hand side of the definition.

Thus $\displaystyle \sup_{i \in I} f_i \left({s}\right)$ is commonly used instead of $\displaystyle \left({\sup_{i \in I} f_i}\right) \left({s}\right)$.


Also see