Complex conjugate vector space

In mathematics, the complex conjugate of a complex vector space  is a complex vector space , which 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 .

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).

Motivation
If  and  are complex vector spaces, a function  is antilinear if

With the use of the conjugate vector space , an antilinear map  can be regarded as an ordinary linear map of type . The linearity is checked by noting:

Conversely, any linear map defined on  gives rise to an antilinear map on .

This is the same underlying principle as in defining opposite ring so that a right -module can be regarded as a left -module, or that of an opposite category so that a contravariant functor   can be regarded as an ordinary functor of type .

Complex conjugation functor
A linear map  gives rise to a corresponding  linear map  which has the same action as . Note that  preserves scalar multiplication because

Thus, complex conjugation  and  define a functor from the category of complex vector spaces to itself.

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 double conjugate  is identical to .

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).

See also
 Linear complex structure

References

Further reading
 Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. . (complex conjugate vector spaces are discussed in section 3.3, pag. 26).

Category:Linear algebra
Vector space