# Definition:Pointwise Scalar Multiplication of Complex-Valued Function

## Definition

Let $S$ be a non-empty set.

Let $f: S \to \C$ be an complex-valued function.

Let $\lambda \in \C$ be an complex number.

Then the pointwise scalar product of $f$ by $\lambda$ is defined as:

$\lambda \times f: S \to \C:$
$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$

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

This can be seen to be an instance of pointwise multiplication where one of the functions is the constant mapping:

$f_\lambda: S \to \C: \map {f_\lambda} s = \lambda$

## Also denoted as

Using the other common notational forms for multiplication, this definition can also be written:

$\forall s \in S: \map {\paren {\lambda \cdot f} } s := \lambda \cdot \map f s$

or:

$\forall s \in S: \map {\paren {\lambda f} } s := \lambda \map f s$