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With conjugation in its first slot, a complex inner product has conjugate symmetry, linearity in its second slot, and positive definiteness. For the proposed Hermitian inner product, direct expansion givesThe last expression vanishes exactly for the zero vector. Conjugate symmetry and linearity also give antilinearity in the first slot: $⟨\alpha z+\beta w,y⟩=\bar{\alpha }⟨z,y⟩+\bar{\beta }⟨w,y⟩$. These verify all three axioms and the associated first-slot consequence.

Use the convention that the Hermitian inner product is conjugate-linear in its first argument and linear in its second. In orthonormal coordinates, $⟨v,w⟩={\sum }_i\overline{v_i}{w}_i$. It satisfies conjugate symmetry, sesquilinearity and positive definiteness. Expanding in the orthonormal basis givesApplying the Hermitian inner product and $⟨e_i,e_j⟩={\delta }_{ij}$ yieldsThis identifies exactly where complex conjugation enters; replacing the first factor by $⟨e_i,v⟩$ would generally be wrong under this convention.