\begin{definition}[Definition:Pointwise Scalar Multiplication of Linear Operators]
Let $X, Y$ be vector spaces.
Let $\map \LL {X,Y}$ denote the set of linear operators on $X$.
Let $F$ be a field.
Then '''pointwise ($F$-)scalar multiplication''' on $\map \LL {X,Y}$ is the binary operation $\cdot: F \times \map \LL {X,Y} \to \map \LL {X,Y}$ defined by:
:$\forall \lambda \in F: \forall T \in \map \LL {X,Y}: \forall x \in X: \map {\paren {\lambda \cdot T} } x := \lambda \cdot \map T x$
where the $\cdot$ on the right is field product.
\end{definition}