# Definition:Pointwise Multiplication of Real-Valued Functions

## Definition

Let $S$ be a non-empty set.

Let $f, g: S \to \R$ be real-valued functions.

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

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

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

Thus pointwise multiplication is seen to be an instance of a pointwise operation on real-valued functions.

## Also denoted as

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

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

or:

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