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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 σ : C → C {\displaystyle \sigma :{\mathbb {C} }\to {\mathbb {C} }\,} , with σ = z ¯ {\displaystyle \sigma ={\bar {z}}} , giving the "canonical" real structure on C {\displaystyle {\mathbb {C} }\,} , that is C = R ⊕ i R {\displaystyle {\mathbb {C} }={\mathbb {R} }\oplus i{\mathbb {R} }\,}.
The conjugation map is antilinear: σ = λ ¯ σ {\displaystyle \sigma ={\bar {\lambda }}\sigma \,} and σ = σ + σ {\displaystyle \sigma =\sigma +\sigma \,}.