Definition:Pointwise Multiplication of Real-Valued Functions

Definition
Let $S$ be a non-empty set. Let $f, g: S \to \Z$ be real-valued functions.

Then the pointwise product of $f$ and $g$ is defined as:
 * $f \times g: S \to \Z:$
 * $\forall s \in S: \left({f \times g}\right) \left({s}\right) := f \left({s}\right) \times g \left({s}\right)$

where the $\times$ on the right hand side is 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:
 * $\forall s \in S: \left({f \cdot g}\right) \left({s}\right) := f \left({s}\right) \cdot g \left({s}\right)$

or:
 * $\forall s \in S: \left({f g}\right) \left({s}\right) := f \left({s}\right) g \left({s}\right)$

Also see

 * Pointwise Multiplication on Real-Valued Functions is Associative
 * Pointwise Multiplication on Real-Valued Functions is Commutative


 * Pointwise Operation on Real-Valued Functions