Definition:Pointwise Addition of Linear Operators
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Definition
Let $V$ be a vector space.
Let $\map \LL V$ denote the set of linear operators on $V$.
- $+: \map \LL V \times \map \LL V \to \map \LL V: \forall S, T \in \map \LL V:$
- $\forall u \in V: \map {\paren {S + T} } u := \map S u + \map T u$
where $+$ on the right hand side is vector addition.
Specific Instances
Specific instantiations of this concept to particular vector spaces are as follows:
Complex Vector Space
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.
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.1$: Continuous and linear maps. Linear transformations