Definition:Pointwise Addition

Definition
Let $S$ be a non-empty set.

Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$.

Let $\mathbb F^S$ be the set of all mappings $f: S \to \mathbb F$.

The (binary) operation of pointwise addition is defined on $\mathbb F^S$ as:


 * $+: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$
 * $\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$

where the $+$ on the is conventional arithmetic addition.

Specific Number Sets
Specific instantiations of this concept to particular number sets are as follows:

Also see

 * Pointwise Addition is Associative
 * Pointwise Addition is Commutative


 * Pointwise Multiplication

It can be seen that these definitions instantiate the more general Pointwise Operation on Number-Valued Functions.