\begin{definition}[Definition:Pointwise Scalar Multiplication of Real-Valued Function]
Let $S$ be a non-empty set.
Let $f: S \to \R$ be an real-valued function.
Let $\lambda \in \R$ be an real number.
Then the '''pointwise scalar product of $f$ by $\lambda$''' is defined as:
:$\lambda \times f: S \to \R:$
::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$
where the $\times $ on the {{RHS}} is real multiplication.
This can be seen to be an instance of pointwise multiplication where one of the functions is the constant mapping:
:$f_\lambda: S \to \R: \map {f_\lambda} s = \lambda$
\end{definition}