Definition:Pointwise Multiplication

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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:

$\times: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$
$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$

where the $\times$ on the right hand side is conventional arithmetic multiplication.


Also denoted as

Using the other common notational forms for multiplication, this definition can also be written:

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

or:

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


Specific Number Sets

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


Integer-Valued Functions

Let $f, g: S \to \Z$ be integer-valued functions.


Then the pointwise product of $f$ and $g$ is defined as:

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

where the $\times$ on the right hand side is integer multiplication.


Rational-Valued Functions

Let $f, g: S \to \Q$ be rational-valued functions.


Then the pointwise product of $f$ and $g$ is defined as:

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

where the $\times$ on the right hand side is rational multiplication.


Real-Valued Functions

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.


Complex-Valued Functions

Let $f, g: S \to \Z$ be complex-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 complex multiplication.


Specific Instance

Pointwise Scalar Multiplication

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:


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

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

where the $\times$ on the right hand side is conventional arithmetic multiplication.


Pointwise Multiplication on Ring of Mappings

Let $\struct {R, +, \circ}$ be a ring.

Let $S$ be a set.

Let $\struct {R^S, +', \circ'}$ be the ring of mappings from $S$ to $R$.


The operation $\circ’$ induced by $\circ$ on the ring of mappings from $S$ to $R$ is called pointwise multiplication and is defined as:

$\forall f, g \in R^S: f \circ’ g \in R^S :$
$\forall s \in S : \map {\paren {f \circ’ g}} x = \map f x \circ \map g x$


Pointwise Multiplication on Ring of Sequences

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {R^\N, +', \circ'}$ be the ring of sequences over $R$.


The operation $\circ’$ induced by $\circ$ on the ring of sequences is called pointwise multiplication and is defined as:

$\forall \sequence {x_n}, \sequence {y_n} \in R^{\N}: \sequence {x_n} \circ' \sequence {y_n} = \sequence {x_n \circ y_n}$


Also see


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