Definition:Pointwise Scalar Multiplication of Linear Operators

Definition

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.

Also see

• Definition:Pointwise Scalar Multiplication of Mappings for pointwise scalar multiplication of more general mappings
• Definition:Pointwise Operation on Real-Valued Functions

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.1$: Continuous and linear maps. Linear transformations