Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 11 / iii / Solution

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.