Unbounded operator

An unbounded operator is a linear map $T:D(T)\subseteq X\to Y$ for which no finite constant $C$ bounds $\Vert Tx{\Vert }_Y\le C\Vert x{\Vert }_X$ on its domain. The domain and norms are therefore part of its specification.

On complex $L^2(0,2\pi )$, take $T=-id/dx$ with periodic $H^1$ domain. The functions $e_n(x)=e^{inx}/\sqrt{2\pi }$ have norm one, while $T{e}_n=n{e}_n$ has norm $|n|$. Hence no such constant exists. Differentiation can nevertheless be bounded when the source is equipped with an $H^1$ norm. An operator's unboundedness thus cannot be inferred from its formula without stating the spaces and domain.