Definition:Pointwise Multiplication
Definition
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:
- $f \times g: S \to \Z:$
- $\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$
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:
- $f \times g: S \to \Q:$
- $\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$
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:
- $f \times g: S \to \R:$
- $\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$
where $\times$ on the right hand side denotes 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:
- $f \times g: S \to \Z:$
- $\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$
where the $\times$ on the right hand side is complex multiplication.
Specific Instances
Pointwise Scalar Multiplication
When one of the functions is the constant mapping $f_\lambda: S \to \mathbb F: \map {f_\lambda} s = \lambda$, the following definition arises:
The (binary) operation of pointwise scalar multiplication is defined on $\mathbb F \times \mathbb F^S$ as:
- $\times: \mathbb F \times \mathbb F^S \to \mathbb F^S: \forall \lambda \in \mathbb F, f \in \mathbb F^S:$
- $\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$
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 pointwise 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 pointwise 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
- Definition:Pointwise Operation on Number-Valued Functions: a more general concept of which these definitions can be seen to be instantiations.