Definition:Pointwise Multiplication of Real-Valued Functions

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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:

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

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:

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


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

Also see