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 ) = z ¯ {\displaystyle \sigma (z)={\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 z ) = λ ¯ σ ( z ) {\displaystyle \sigma (\lambda z)={\bar {\lambda }}\sigma (z)\,} and σ ( z 1 + z 2 ) = σ ( z 1 ) + σ ( z 2 ) {\displaystyle \sigma (z_{1}+z_{2})=\sigma (z_{1})+\sigma (z_{2})\,} .
Vector space A real structure on a complex vector space V is an antilinear involution σ : V → V {\displaystyle \sigma :V\to V} . A real structure defines a real subspace V R ⊂ V {\displaystyle V_{\mathbb {R} }\subset V} , its fixed locus, and the natural map
V R ⊗ R C → V {\displaystyle V_{\mathbb {R} }\otimes _{\mathbb {R} }{\mathbb {C} }\to V}
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 t ∈ V {\displaystyle t\in V\,} and t ≠ 0 {\displaystyle t\neq 0} then the vectors t {\displaystyle t\,} and i t {\displaystyle it\,} are linearly independent in the realification of V. Hence:
dim R V = 2 dim C V {\displaystyle \dim _{\mathbb {R} }V=2\dim _{\mathbb {C} }V}
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 σ : V → V {\displaystyle \sigma :V\to V\,} be an antilinear map such that σ ∘ σ = i d V {\displaystyle \sigma \circ \sigma =id_{V}\,} , that is an antilinear involution of the complex space V. Any vector v ∈ V {\displaystyle v\in V\,} can be written v = v + + v − {\displaystyle {v=v^{+}+v^{-}}\,} , where v + = 1 2 ( v + σ v ) {\displaystyle v^{+}={1 \over {2}}(v+\sigma v)} and v − = 1 2 ( v − σ v ) {\displaystyle v^{-}={1 \over {2}}(v-\sigma v)\,} . Therefore, one gets a direct sum of vector spaces V = V + ⊕ V − {\displaystyle V=V^{+}\oplus V^{-}\,} where:
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