# Definition:Pointwise Supremum of Real-Valued Functions

## Definition

Let $S$ be a set.

Let $\family {f_i}_{i \mathop \in I}, f_i: S \to \R$ be an $I$-indexed collection of real-valued functions.

Then the **pointwise supremum of $\family {f_i}_{i \mathop \in I}$**, denoted $\ds \sup_{i \mathop \in I} f_i: S \to \overline \R$, is defined by:

- $\ds \map {\paren {\sup_{i \mathop \in I} f_i} } s := \sup_{i \mathop \in I} \map {f_i} s$

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

Thus $\ds \sup_{i \mathop \in I} \map {f_i} s$ is commonly used instead of $\ds \map {\paren {\sup_{i \mathop \in I} f_i} } s$.

## Also see

- Definition:Pointwise Supremum of Extended Real-Valued Functions, an generalization to extended real-valued functions
- Definition:Pointwise Supremum, a further generalization replacing $\R$ with a general ordered set $T$