Definition:Pointwise Operation on Complex-Valued Functions

Definition
Let $S$ be a non-empty set. Let $\C^S$ be the set of all mappings $f: S \to \C$, where $\C$ is the set of complex numbers.

Let $\oplus$ be a binary operation on $\C$.

Define $\oplus: \C^S \times \C^S \to \C^S$, called pointwise $\oplus$, by:


 * $\forall f, g \in \C^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$

In the above expression, the operator on the is the given $\oplus$ on the complex numbers.

Specific Instantiations
When $\oplus$ has a specific name, it is usual to name the corresponding pointwise operation by prepending pointwise to that name:

Also defined as
Sometimes an operation cannot be consistently defined on all of $\C$.

Often one then still speaks about a pointwise operation by suitably restricting above definition, adapting it wherever necessary.

Examples of such suitably restricted pointwise operations are listed under Partial Examples below.

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


 * Definition:Pointwise Scalar Multiplication of Complex-Valued Functions
 * Definition:Pointwise Maximum of Complex-Valued Functions
 * Definition:Pointwise Minimum of Complex-Valued Functions
 * Definition:Absolute Value of Complex-Valued Function