Definition:Densely-Defined Linear Operator

Definition
Let $\Bbb F \in \set {\R, \C}$.

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\Bbb F$.

Let $\map D T$ be a everywhere dense linear subspace of $\HH$.

Let $T : \map D T \to \HH$ be a function such that:


 * $\map T {\lambda x + \mu y} = \lambda T x + \mu T y$ for all $\lambda, \mu \in \Bbb F$ and $x, y \in \map D T$.

Then we say that $\struct {\map D T, T}$ is a densely-defined linear operator.