Definition

The set of all linear maps from $V\to W$ is defined as $$\mathscr{L}(V,W):=\{T:V\to W|T\text{ is a linear map} \}$$ If $V=W$ then we shorten this notation to $\mathscr{L}(V)$. An element of $\mathscr{L}(V)$ is called an operator.