Definition:Pointwise Operation on Number-Valued Functions

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 $f: S \to \mathbb F$.

Let $\oplus$ be a binary operation on $\mathbb F$.

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


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

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

Specific Operations
This concept often occurs when $\oplus$ is a conventional arithmetic operation, for example addition or multiplication.

In this case it is usual to refer the corresponding pointwise operation by prepending pointwise to that name, so as to obtain pointwise addition and pointwise multiplication.

Also see

 * Definition:Pointwise Operation, where it is shown that this concept can be applied where the codomain can be any algebraic structure, not just $\Z, \Q, \R$ or $\C$.