Definition:Pointwise Operation

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Let $S$ be a set.

Let $\struct {T, \circ}$ be an algebraic structure.

Let $T^S$ be the set of all mappings from $S$ to $T$.

Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings.

Then the operation $f \oplus g$ is defined on $T^S$ as follows:

$f \oplus g: S \to T: \forall x \in S: \map {\paren {f \oplus g} } x = \map f x \circ \map g x$

The operation $\oplus$ is called the pointwise operation on $T^S$ induced by $\circ$.

Induced Structure

The algebraic structure $\struct {T^S, \oplus}$ is called the algebraic structure on $T^S$ induced by $\circ$.

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.

Also known as

A pointwise operation is also referred to as an induced operation.

It is usual to use the same symbol for the pointwise operation as for the operation that induces it.

Thus one would refer to the structure on $T^S$ induced by $\circ$ as $\struct {T^S, \circ}$.

In most reference works, the precise properties of a pointwise operation are taken to be implicitly inherited from its base operation.


Cube and Sine Functions

Let $f$ and $g$ be the real functions defined as

\(\ds \forall x \in \R: \, \) \(\ds \map f x\) \(=\) \(\ds x^3\)
\(\ds \forall x \in \R: \, \) \(\ds \map g x\) \(=\) \(\ds \sin x\)

Then for all $x \in \R$:

\(\ds \map {\paren {f + g} } x\) \(=\) \(\ds x^3 + \sin x\)
\(\ds \map {\paren {f - g} } x\) \(=\) \(\ds x^3 - \sin x\)
\(\ds \map {\paren {f \times g} } x = \map {\paren {g \times f} } x\) \(=\) \(\ds x^3 \sin x\)
\(\ds \map {\paren {f \times f} } x\) \(=\) \(\ds x^6\)

Contrast this with:

\(\ds \map {\paren {f \circ g} } x\) \(=\) \(\ds \paren {\sin x}^3\)
\(\ds \map {\paren {g \circ f} } x\) \(=\) \(\ds \map \sin {x^3}\)
\(\ds \map {\paren {f \circ f} } x\) \(=\) \(\ds x^9\)

Also see