The Hilbert Space is an inner product space, which is complete with respect to the norm defined by the inner product.

See also on Quantiki

• $U^\dagger$ is the adjoint of $U$, and in matrix representation is given by $(U^*)^T$, where $^*$ marks the complex conjugate and $^T$ the transpose.
• An operator (matrix) $U$ is said to be normal if $U^\dagger U = U U^\dagger$.
  • A normal operator is diagonizable (spectral decomposition)
• An operator (matrix) $U$ is said to be unitary if $U^\dagger U = I$.
• An operator (matrix) $U$ is said to be hermitian if $U = U^\dagger$.
  • A hermitian operator is normal.