between two complex vector spaces is said to be antilinear or conjugate-linear if
If the vector spaces are real then antilinearity is the same as linearity.
Antilinear maps occur in quantum mechanics in the study of time reversal and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices.
Scalar-valued antilinear maps often arise when dealing with complex inner products and Hilbert spaces.
A function is called antilinear or conjugate linear if it is additive and conjugate homogeneous.
An antilinear functional on a vector space
is a scalar-valued antilinear map.
In contrast, a linear map is a function that is additive and homogeneous, where
{\displaystyle f(ax)=af(x)\quad {\text{ for all vectors }}x{\text{ and all scalars }}a.}
may be equivalently described in terms of the linear map
to the complex conjugate vector space
of rank 1, we can construct an anti-linear dual map which is an anti-linear map
for some fixed real numbers
We can extend this to any finite dimensional complex vector space, where if we write out the standard basis
The anti-linear dual[1]pg 36 of a complex vector space
is a special example because it is isomorphic to the real dual of the underlying real vector space of
In the other direction, there is the inverse map sending a real dual vector
giving the desired map.
The class of semilinear maps generalizes the class of antilinear maps.
is called the algebraic anti-dual space of
is a topological vector space, then the vector space of all continuous antilinear functionals on
is called the continuous anti-dual space or simply the anti-dual space of
is a normed space then the canonical norm on the (continuous) anti-dual space
This formula is identical to the formula for the dual norm on the continuous dual space
Canonical isometry between the dual and anti-dual The complex conjugate
This says exactly that the canonical antilinear bijection defined by
reduces down to the identity map.
is an inner product space then both the canonical norm on
satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on
) is consistent with the dual norm (that is, as defined above by the supremum over the unit ball); explicitly, this means that the following holds for every