Definition:Pointwise Scalar Multiplication of Mappings

Definition

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

Let $\struct {S, \circ}_R$ be an $R$-algebraic structure.

Let $X$ be a non-empty set.

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

Then pointwise ($R$)-scalar multiplication on $S^X$ is the binary operation $\circ: R \times S^X \to S^X$ (the $\circ$ is the same as for $S$) defined by:

$\forall \lambda \in R: \forall f \in S^X: \forall x \in X: \map {\paren {\lambda \circ f} } x := \lambda \circ \map f x$

The double use of $\circ$ is justified as $\struct {S^X, \circ}_R$ inherits all abstract-algebraic properties $\struct {S, \circ}_R$ might have.

This is rigorously formulated and proved on Mappings to R-Algebraic Structure form Similar R-Algebraic Structure.

Examples

• Definition:Pointwise Scalar Multiplication of Real-Valued Functions

Also see

• Definition:Pointwise Scalar Multiplication of Extended Real-Valued Functions: not an example as the extended real numbers are not a ring.
• Definition:Pointwise Addition of Mappings: a similar concept; the two are often used in conjunction in the context of vector spaces.