Adjoint of a densely defined operator ($⟨Tf,h⟩=⟨f,T^{\ast }h⟩(f\in D(T))$)

For a densely defined operator on a Hilbert space, $h$ belongs to $D(T^{\ast })$ exactly when the functional $f\mapsto ⟨Tf,h⟩$ is bounded in the ambient Hilbert norm on $D(T)$. Density makes its representing vector $T^{\ast }h$ unique. Differential expressions alone do not determine this domain.

// Target: analysis.bigb