Definition 2.1.1.label Let $H$ be a Hilbert space, $\varphi: H^{*} \to H$ be the canonical map, and $T: D(T) \to H$ be a densely defined linear operator. Let
\[D(T^{*}) = \bracsn{\psi \in H| \dpb{T\cdot, \psi}{H} \in H^*}\]
then the adjoint of $T$ is the mapping
\[D(T^{*}) \to H \quad \phi \mapsto \varphi(\dpb{T\cdot, \psi}{H})\]