Definition:Pointwise Scalar Multiplication of Extended Real-Valued Functions

Definition
Let $S$ be a set, and let $f : S \to \overline{\R}$ be an extended real-valued function.

Let $\lambda \in \overline{\R}$.

Then $\lambda \cdot f: S \to \overline{\R}$, the pointwise $\lambda$-multiple of $f$ is defined by (for all $s \in S$):


 * $\left({\lambda \cdot f}\right) \left({s}\right) := \lambda \cdot f \left({s}\right)$

where the $\cdot$ on the right is extended real multiplication.

Pointwise scalar multiplication thence is an instance of a pointwise operation on extended real-valued functions.

Also see

 * Pointwise Scalar Multiplication of Extended Real-Valued Functions is Associative
 * Pointwise Scalar Multiplication of Mappings for pointwise scalar multiplication of more general mappings
 * Pointwise Scalar Multiplication of Real-Valued Functions
 * Pointwise Operation on Extended Real-Valued Functions