Definition:Pointwise Supremum of Real-Valued Functions

From ProofWiki
Jump to navigation Jump to search

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 $\displaystyle \sup_{i \mathop \in I} f_i: S \to \overline \R$, is defined by:

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

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

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


Also see