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

Also see

 * Definition:Pointwise Addition of Complex-Valued Functions
 * Definition:Pointwise Multiplication of Complex-Valued Functions


 * Definition:Pointwise Scalar Multiplication of Number-Valued Function: a more general concept of which this is a specific instance