The main example is the case that the real vector space V is a hermitian vector space (V, h), i.e., V is equipped with a complex structure J that is an orthogonal transformation with respect to the inner product g on V. Then V ⊗ ⊗ R C splits in the ± i eigenspaces of J. These eigenspaces are isotropic for the complexification of g and can be identified with the complex vector space (V, J) and its complex conjugate (V, − J). Therefore, for a hermitian vector space (V, h) the vector space Λ Λ C ⋅ ⋅ V ¯ ¯ (as well as its complex conjugate Λ Λ C ⋅ ⋅ V is a spinor space for the underlying real euclidean vector space.