= Solution
Adopt the convention that an <inner product> is linear in its second slot. Over $\mathbb R$, it is a <symmetric bilinear form> with $\langle v,v\rangle\geq0$, equality only for $v=0$. Over $\mathbb C$, a <Hermitian inner product> obeys
$$
\langle u,av+bw\rangle=a\langle u,v\rangle+b\langle u,w\rangle,\qquad
\langle u,v\rangle=\overline{\langle v,u\rangle},\qquad
\langle v,v\rangle>0\quad(v\ne0).
$$
These imply conjugate linearity in the first slot: $\langle au+bv,w\rangle=\overline a\langle u,w\rangle+\overline b\langle v,w\rangle$. The strict positivity condition distinguishes an <inner product> from a general <Hermitian form>.
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