Real structure

In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation map formula_1, with formula_2, giving the "canonical" real structure on formula_3, that is formula_4. 
The conjugation map is antilinear: formula_5 and formula_6.
Vector space.
A real structure on a complex vector space "V" is an antilinear involution formula_7. A real structure defines a real subspace formula_8, its fixed locus, and the natural map
is an isomorphism. Conversely any vector space that is the complexification
of a real vector space has a natural real structure.
One first notes that every complex space "V" has a realification obtained by taking the same vectors as in the original set and restricting the scalars to be real. If formula_10 and formula_11 then the vectors formula_12 and formula_13 are linearly independent in the realification of "V". Hence:
Naturally, one would wish to represent "V" as the direct sum of two real vector spaces, the "real and imaginary parts of "V"". There is no canonical way of doing this: such a splitting is an additional real structure in "V". It may be introduced as follows. Let formula_15 be an antilinear map such that formula_16, that is an antilinear involution of the complex space "V". 
Any vector formula_17 can be written formula_18, 
where formula_19 and formula_20.
Therefore, one gets a direct sum of vector spaces formula_21 where:
Both sets formula_24 and formula_25 are real vector spaces. The linear map formula_26, where formula_27, is an isomorphism of real vector spaces, whence:
The first factor formula_24 is also denoted by formula_30 and is left invariant by formula_31, that is formula_32. The second factor formula_25 is 
usually denoted by formula_34. The direct sum formula_21 reads now as:
i.e. as the direct sum of the "real" formula_30 and "imaginary" formula_34 parts of "V". This construction strongly depends on the choice of an antilinear involution of the complex vector space "V". The complexification of the real vector space formula_30, i.e., 
formula_40 admits 
a natural real structure and hence is canonically isomorphic to the direct sum of two copies of formula_41:
It follows a natural linear isomorphism formula_43 between complex vector spaces with a given real structure.
A real structure on a complex vector space "V", that is an antilinear involution formula_15, may be equivalently described in terms of the linear map formula_45 from the vector space formula_46 to the complex conjugate vector space formula_47 defined by
Algebraic variety.
For an algebraic variety defined over a subfield of the real numbers,
the real structure is the complex conjugation acting on the points of the variety in complex projective or affine space.
Its fixed locus is the space of real points of the variety (which may be empty).
Scheme.
For a scheme defined over a subfield of the real numbers, complex conjugation
is in a natural way a member of the Galois group of the algebraic closure of the basefield.
The real structure is the Galois action of this conjugation on the extension of the
scheme over the algebraic closure of the base field.
The real points are the points whose residue field is fixed (which may be empty).