In mathematics, the complex conjugate of a complex vector space is a complex vector space that has the same elements and additive group structure as but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of satisfies
where is the scalar multiplication of and is the scalar multiplication of
The letter stands for a vector in is a complex number, and denotes the complex conjugate of [1]
More concretely, the complex conjugate vector space is the same underlying real vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure (different multiplication by ).
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.[citation needed]
Thus, the complex conjugate to a vector particularly in finite dimension case, may be denoted as (v-dagger, a row vector that 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).
- Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. ISBN 0-387-19078-3. (complex conjugate vector spaces are discussed in section 3.3, pag. 26).