Definition:Pointwise Scalar Multiplication of Mappings

Definition
Let $\left({R, +_R, \times_R}\right)$ be a ring, and let $\left({S, \circ}\right)_R$ be an $R$-algebraic structure.

Let $X$ be a nonempty set, and 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:


 * $\forall \lambda \in R: \forall f \in S^X: \forall x \in X: \left({\lambda \circ f}\right) \left({x}\right) := \lambda \circ f \left({x}\right)$

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

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

Examples

 * Pointwise Scalar Multiplication of Real-Valued Functions

Also see

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