# Definition:Pointwise Scalar Multiplication of Mappings/Real-Valued Functions

## Definition

Let $S$ be a non-empty set, and let $\R^S$ be the set of all mappings $f: S \to \R$.

Then pointwise ($\R$-)scalar multiplication on $\R^S$ is the binary operation $\cdot: \R \times \R^S \to \R^S$ defined by:

$\forall \lambda \in \R: \forall f \in \R^S: \forall s \in S: \left({\lambda \cdot f}\right) \left({s}\right) := \lambda \cdot f \left({s}\right)$

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

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

## Also known as

Like the $\cdot$ for real multiplication, its pointwise analog (also denoted $\cdot$) is often omitted.

That is, one often encounters $\lambda f$ instead of $\lambda \cdot f$.

Furthermore, by the way pointwise $\R$-scalar multiplication is defined, one often disposes of parentheses.

Thus one simply writes $\lambda f \left({s}\right)$, leaving unspecified whether this means $\left({\lambda \cdot f}\right) \left({s}\right)$ or $\lambda \cdot f \left({s}\right)$.

This is justified as the expressions are equal in any case, and it saves one from writing excessive parentheses.