Definition:Pointwise Multiplication

Definition
Let $S$ be a non-empty set.

Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$.

Let $\mathbb F^S$ be the set of all mappings $f: S \to \mathbb F$.

The (binary) operation of pointwise multiplication is defined on $\mathbb F^S$ as:


 * $\times: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$
 * $\forall s \in S: \left({f \times g}\right) \left({s}\right) := f \left({s}\right) \times g \left({s}\right)$

where the $\times$ on the RHS is conventional arithmetic multiplication.

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

Specific Number Sets
Specific instantiations of this concept to particular number sets are as follows:

Specific Instance
When one of the functions is the constant mapping $f_\lambda: S \to \mathbb F: f_\lambda \left({s}\right) = \lambda$, the following definition arises:

Also see

 * Pointwise Multiplication is Associative
 * Pointwise Multiplication is Commutative


 * Pointwise Addition

It can be seen that these definitions instantiate the more general Pointwise Operation on Number-Valued Functions.