Definition:Pointwise Supremum of Real-Valued Functions

Definition

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

However, mind that this pointwise supremum need not be a real-valued function.

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