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).
Read more about this topic: Complex Conjugate Vector Space
Famous quotes containing the words complex and/or space:
“By “object” is meant some element in the complex whole that is defined in abstraction from the whole of which it is a distinction.”
—John Dewey (1859–1952)
“In bourgeois society, the French and the industrial revolution transformed the authorization of political space. The political revolution put an end to the formalized hierarchy of the ancien regimé.... Concurrently, the industrial revolution subverted the social hierarchy upon which the old political space was based. It transformed the experience of society from one of vertical hierarchy to one of horizontal class stratification.”
—Donald M. Lowe, U.S. historian, educator. History of Bourgeois Perception, ch. 4, University of Chicago Press (1982)