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
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The conjugation map is antilinear:
Vector space
A real structure on a complex vector space V is an antilinear involution
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 real form obtained by taking the same vectors as in the original set and restricting the scalars to be real. If
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
Therefore, one gets a direct sum of vector spaces
Both sets
The first factor
i.e. as the direct sum of the "real"
It follows a natural linear isomorphism
A real structure on a complex vector space V, that is an antilinear involution
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).