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

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## 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:

- $\forall \lambda \in \R: \forall f \in \R^S: \forall s \in S: \map {\paren {\lambda \cdot f} } s := \lambda \cdot \map f s$

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 $\lambda \map f s$, leaving unspecified whether this means $\map {\paren {\lambda \cdot f} } s$ or $\lambda \cdot \map f s$.

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

## Also see

- Pointwise Scalar Multiplication of Real-Valued Functions is Associative
- Definition:Pointwise Scalar Multiplication of Mappings for
**pointwise scalar multiplication**of more general mappings

## Sources

- 1961: I.M. Gel'fand:
*Lectures on Linear Algebra*(2nd ed.) ... (previous) ... (next): $\S 1$: $n$-Dimensional vector spaces