Definition:Pointwise Addition on Complex Vector Space

Definition

Let $\C^n$ be a complex vector space.

Let $S$ and $T$ be linear operators on $\C^n$.

Then the pointwise sum of $S$ and $T$ is defined as:

$S + T: \C^n \to \C^n:$

$\forall u \in \C^n: \map {\paren {S + T} } u := \map S u + \map T u$

where $+$ on the right hand side is complex vector addition.

Thus pointwise addition on $\C^n$ is seen to be an instance of a pointwise addition of linear operators.

Sources

• 1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur: Quantum Mechanics ... (previous) ... (next): Chapter $2$: Mathematical Background: $2.3$ Linear Operators and Matrices