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 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:

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.