# Definition:Pointwise Operation on Complex-Valued Functions

## Definition

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 right hand side 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:

### Pointwise Addition

Let $f, g: S \to \C$ be complex-valued functions.

Then the **pointwise sum of $f$ and $g$** is defined as:

- $f + g: S \to \C:$
- $\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$

where $+$ on the right hand side is complex addition.

### Pointwise Multiplication

Let $f, g: S \to \Z$ be complex-valued functions.

Then the **pointwise product of $f$ and $g$** is defined as:

- $f \times g: S \to \Z:$
- $\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$

where the $\times$ on the right hand side is complex multiplication.

## 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.