Definition:Pointwise Operation on Complex-Valued Functions

Definition
Let $S$ be a 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:


 * $\forall f, g \in \C^S: \forall s \in S: \left({f \oplus g}\right) \left({s}\right) := f \left({s}\right) \oplus g \left({s}\right)$

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

Specific Instantiations
When $\oplus$ has a specific name, for example "addition" or "multiplication", 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

 * Pointwise Scalar Multiplication
 * Pointwise Maximum
 * Pointwise Minimum
 * Absolute Value


 * Pointwise Limit

It can be seen that these definitions instantiate the general induced operation on set of mappings.