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

Also see

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


 * Definition:Pointwise Operation on Real-Valued Functions