 Solution

Adopt the convention that an inner product is linear in its second slot. Over $\mathbb{R}$, it is a symmetric bilinear form with $⟨v,v⟩\ge 0$, equality only for $v=0$. Over $\mathbb{C}$, a Hermitian inner product obeysThese imply conjugate linearity in the first slot: $⟨au+bv,w⟩=\overline{a}⟨u,w⟩+\overline{b}⟨v,w⟩$. The strict positivity condition distinguishes an inner product from a general Hermitian form.