Unitary means we have a complex (real) vector space with a complex (sesquilinear / real: bilinear) inner product, which is also called a pre-Hilbert space or inner product space. If it is of finite dimension, then we the inner product defines an isometric isomorphism on ##\mathbb{C}^n## (resp. ##\mathbb{R}^n\;##) which is complete.kent davidge said:In texts treating Hilbert spaces, it's usually given as an example that "any finite dimensional unitary space is complete", but I've found no proof so far and failed prove it myself.