Reality structure

In mathematics, a reality structure on a complex vector space "V" is a decomposition of "V" into two real subspaces, called the real and imaginary parts of "V":
Here "V"R is a real subspace of "V", i.e. a subspace of "V" considered as a vector space over the real numbers. If "V" has complex dimension "n" (real dimension 2"n"), then "V"R must have real dimension "n". 
The standard reality structure on the vector space formula_2 is the decomposition
In the presence of a reality structure, every vector in "V" has a real part and an imaginary part, each of which is a vector in "V"R:
In this case, the complex conjugate of a vector "v" is defined as follows:
This map formula_6 is an antilinear involution, i.e.
Conversely, given an antilinear involution formula_8 on a complex vector space "V", it is possible to define a reality structure on "V" as follows. Let
and define
Then
This is actually the decomposition of "V" as the eigenspaces of the real linear operator "c". The eigenvalues of "c" are +1 and −1, with eigenspaces "V"R and formula_12 "V"R, respectively. Typically, the operator "c" itself, rather than the eigenspace decomposition it entails, is referred to as the reality structure on "V".