Definition:Pointwise Operation on Number-Valued Functions

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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 $f: S \to \mathbb F$.


Let $\oplus$ be a binary operation on $\mathbb F$.

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

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


Specific Number Sets

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


Integer-Valued Functions

Let $\Z^S$ be the set of all mappings $f: S \to \Z$, where $\Z$ is the set of integers.


Let $\oplus$ be a binary operation on $\Z$.

Define $\oplus: \Z^S \times \Z^S \to \Z^S$, called pointwise $\oplus$, by:

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

In the above expression, the operator on the RHS is the given $\oplus$ on the integers.


Rational-Valued Functions

Let $\Q^S$ be the set of all mappings $f: S \to \Q$, where $\Q$ is the set of rational numbers.


Let $\oplus$ be a binary operation on $\Q$.

Define $\oplus: \Q^S \times \Q^S \to \Q^S$, called pointwise $\oplus$, by:

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

In the above expression, the operator on the RHS is the given $\oplus$ on the rational numbers.


Real-Valued Functions

Let $\R^S$ be the set of all mappings $f: S \to \R$, where $\R$ is the set of real numbers.


Let $\oplus$ be a binary operation on $\R$.

Define $\oplus: \R^S \times \R^S \to \R^S$, called pointwise $\oplus$, by:

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

In the above expression, the operator on the right hand side is the given $\oplus$ on the real numbers.


Complex-Valued Functions

Let $\C^S$ be the set of all mappings $f: S \to \C$, where $\C$ is the set of complex numbers.


Let $\oplus$ be a binary operation on $\C$.

Define $\oplus: \C^S \times \C^S \to \C^S$, called pointwise $\oplus$, by:

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

In the above expression, the operator on the RHS is the given $\oplus$ on the complex numbers.


Specific Operations

This concept often occurs when $\oplus$ is a conventional arithmetic operation, for example addition or multiplication.

In this case it is usual to refer the corresponding pointwise operation by prepending pointwise to that name, so as to obtain pointwise addition and pointwise multiplication.

Pointwise Addition

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

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

where the $+$ on the right hand side is conventional arithmetic addition.


Pointwise Multiplication

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 see