Unbounded operator
= Unbounded operator
An <unbounded operator> is a <linear map> $T:D(T)\subseteq X\to Y$ for which no finite constant $C$ bounds $\|Tx\|_Y\leq C\|x\|_X$ on its domain. The domain and norms are therefore part of its specification.
On complex $L^2(0,2\pi)$, take $T=-i\,d/dx$ with periodic $H^1$ domain. The <functions> $e_n(x)=e^{inx}/\sqrt{2\pi}$ have <norm> one, while $Te_n=ne_n$ 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.