[[Endomorphism ring]]
# Summable endomorphisms

Let $V$ be a [[vector space]] over $\mathbb{K}$ and $\{ x_{i} \}_{i \in I} \sube \End V$ be a family of linear operators.
We say$\{ x_{i} \}_{i \in I}$ is **summable** iff for any $v \in V$, $x_{(-)}v$ has [[Support of a map|finite support]], #m/def/linalg 
and thence[^1988]
$$
\begin{align*}
\sum_{i \in I}x_{i} : V &\to V \\
v &\mapsto \sum_{i \in I} x_{i}v
\end{align*}
$$

  [^1988]: 1988\. [[Sources/@frenkelVertexOperatorAlgebras1988|Vertex operator algebras and the Monster]], §2.1, p. 49

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