Definition:Pointwise Addition on Complex Vector Space

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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.