Definition:Pointwise Multiplication of Integer-Valued Functions
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Definition
Let $f, g: S \to \Z$ be integer-valued functions.
Then the pointwise product of $f$ and $g$ is defined as:
- $f \times g: S \to \Z:$
- $\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$
where the $\times$ on the right hand side is integer multiplication.
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 Integer-Valued Functions is Associative
- Pointwise Multiplication on Integer-Valued Functions is Commutative
- Pointwise multiplication is seen to be an instance of a pointwise operation on integer-valued functions.