Definition:Pointwise Operation
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Definition
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.
Examples
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
- Definition:Pointwise Operation on Number-Valued Functions: this definition crystallises when $T$ is taken to be one of the standard number sets $\N, \Z, \Q, \R$ and $\C$.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces