Complex conjugate vector space
In mathematics, the (formal) complex conjugate of a complex vector space is the complex vector space consisting of all formal complex conjugates of elements of . That is, is a vector space whose elements are in one-to-one correspondence with the elements of :
Here and are vectors in , is a complex number, and denotes the complex conjugate of .
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 J (different multiplication by i).
If and are complex vector spaces, a function is antilinear if
for all and .
One reason to consider the vector space is that it makes antilinear maps into linear maps. Specifically, if is an antilinear map, then the corresponding map defined by
is linear. Conversely, any linear map defined on gives rise to an antilinear map on .
One way of thinking about this correspondence is that the map defined by
is an antilinear bijection. Thus if is linear, then composition is antilinear, and vice versa.
Conjugate linear maps
Any linear map induces a conjugate linear map , defined by the formula
The conjugate linear map is linear. Moreover, the identity map on induces the identity map , and
If and are finite-dimensional and the map is described by the complex matrix with respect to the bases of and of , then the map is described by the complex conjugate of with respect to the bases of and of .
Structure of the conjugate
The vector spaces and have the same dimension over the complex numbers and are therefore isomorphic as complex vector spaces. However, there is no natural isomorphism from to . (The map is not an isomorphism, since it is antilinear.)
The double conjugate is naturally isomorphic to , with the isomorphism defined by
Usually the double conjugate of is simply identified with .
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).
- Budinich, P. and Trautman, A. The Spinorial Chessboard. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (complex conjugate vector spaces are discussed in section 3.3, pag. 26).