Let $V$ be a vector space over $\mathbb{C}$ equipped with a mapping $\langle , \rangle \: V \times V \to \mathbb{C}$ such that
$\langle , \rangle$ is linear in the first argument and conjugate linear in the second argument.
$\langle , \rangle$ is positive definite in that $\langle v , v\rangle \geq 0$ for all $v \in V$ with $\langle v , v\rangle=0$ iff $v=0$.
Does the above two assumptions make $\langle , \rangle$ a genuine inner product? That is, conjugate symmetry $\langle v , w\rangle=\overline{\langle w , v\rangle}$ is implied by the above two assumptions?
It is always confusing...Could anyone please clarify?