# Definition An operator $\mathcal{O}$ on a [[vector space]] $V$ is a [[map]] from the vector space to itself $ \mathcal{O}:V\rightarrow V $ Given $v \in V$, the proper notation is $\mathcal{O}(v)$, but it is often abbreviated $\mathcal{O} v$. Mathematically speaking, an operator is an [[Endomorphism|endomorphism]].