Adjoint of an unbounded operator

Summary

The adjoint of a densely defined operator $T$ between Hilbert spaces is an operator $T^*$ in the opposite direction. Its domain consists of the vectors $y$ for which $x \mapsto \ip{Tx}{y}$ is bounded on the domain of $T$. It extends the Hilbert space adjoint of bounded operators, and the domain of $T^*$ is part of the answer. The adjoint is needed to define self-adjoint unbounded operators, the observables of quantum mechanics.