Complex conjugate vector space

In mathematics, the complex conjugate of a complex vector space formula_1 is a complex vector space formula_2, which has the same elements and additive group structure as formula_3, but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of formula_2 satisfies
where formula_6 is the scalar multiplication of formula_2 and formula_8 is the scalar multiplication of formula_3.
The letter formula_10 stands for a vector in formula_1, formula_12 is a complex number, and formula_13 denotes the complex conjugate of formula_12.
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 formula_1 and formula_16 are complex vector spaces, a function formula_17 is antilinear if
With the use of the conjugate vector space formula_2, an antilinear map formula_20 can be regarded as an ordinary linear map of type formula_21. The linearity is checked by noting:
Conversely, any linear map defined on formula_2 gives rise to an antilinear map on formula_1.
This is the same underlying principle as in defining opposite ring so that a right formula_25-module can be regarded as a left formula_26-module, or that of an opposite category so that a contravariant functor formula_27 can be regarded as an ordinary functor of type formula_28.
Complex conjugation functor.
A linear map formula_29 gives rise to a corresponding linear map formula_30 which has the same action as formula_31. Note that formula_32 preserves scalar multiplication because
Thus, complex conjugation formula_34 and formula_35 define a functor from the category of complex vector spaces to itself.
If formula_1 and formula_16 are finite-dimensional and the map formula_38 is described by the complex matrix formula_39 with respect to the bases formula_40 of formula_1 and formula_42 of formula_16, then the map formula_32 is described by the complex conjugate of formula_39 with respect to the bases formula_46 of formula_2 and formula_48 of formula_49.
Structure of the conjugate.
The vector spaces formula_1 and formula_2 have the same dimension over the complex numbers and are therefore isomorphic as complex vector spaces. However, there is no natural isomorphism from formula_1 to formula_2.
The double conjugate formula_54 is identical to formula_3.
Complex conjugate of a Hilbert space.
Given a Hilbert space formula_56 (either finite or infinite dimensional), its complex conjugate formula_57 is the same vector space as its continuous dual space formula_58.
There is one-to-one antilinear correspondence between continuous linear functionals and vectors.
In other words, any continuous linear functional on formula_56 is an inner multiplication to some fixed vector, and vice versa.
Thus, the complex conjugate to a vector formula_60, particularly in finite dimension case, may be denoted as formula_61 (v-star, a row vector which is the conjugate transpose to a column vector formula_60).
In quantum mechanics, the conjugate to a "ket vector" formula_63 is denoted as formula_64 – a "bra vector" (see bra–ket notation).