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
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
are complex vector spaces, a function
With the use of the conjugate vector space
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 the opposite ring so that a right
-module, or that of an opposite category so that a contravariant functor
can be regarded as an ordinary functor of type
gives rise to a corresponding linear map
preserves scalar multiplication because
define a functor from the category of complex vector spaces to itself.
have the same dimension over the complex numbers and are therefore isomorphic as complex vector spaces.
(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
– a bra vector (see bra–ket notation).