Complex Conjugate Vector Space - Complex Conjugate of A Hilbert Space

Complex Conjugate of A Hilbert Space

Given a Hilbert space (either finite or infinite dimensional), its complex conjugate is the same vector space as its continuous dual space . There is one-to-one antilinear correspondence between continuous linear functionals and vectors. In other words, any continuous linear functional on is an inner multiplication to some fixed vector, and vice versa.

Thus, the complex conjugate to a vector, particularly in finite dimension case, may be denoted as (v-star, a row vector which is the conjugate transpose to a column vector ). In quantum mechanics, the conjugate to a ket vector is denoted as – a bra vector (see bra-ket notation).