The Bogoliubov inner product satisfies all the axioms of the inner product: it is sesquilinear, positive semidefinite (i.e., ⟨ ⟨ X , X ⟩ ⟩ A ≥ ≥ 0 ), and satisfies the symmetry property ⟨ ⟨ X , Y ⟩ ⟩ A = ( ⟨ ⟨ Y , X ⟩ ⟩ A ) ∗ ∗ where α α ∗ ∗ is the complex conjugate of α α .