Hilbert space adjoint

Summary

The adjoint of a bounded operator $T \colon H \to K$ between Hilbert spaces is the bounded operator $T^* \colon K \to H$ characterised by $\ip{Tx}{y} = \ip{x}{T^* y}$. It extends the adjoint of a linear map between finite-dimensional inner product spaces, and the conjugate transpose of matrices. The adjoint is used to define self-adjoint, normal, unitary and positive operators, and it makes the bounded operators on a Hilbert space a C*-algebra.