Let $V$ be a finite dimensional complex inner product space. Let $J$ be a conjugate linear map from $V$ to $V$ such that $J^2 =1$.
Can we say $\langle Ju, Jv \rangle = \langle v, u \rangle$ for all $u, v$ in $V$?
Conjugate linear means $J(u+v) = J(u) + J(v)$ and $J(a.v) = \overline{a}.J(v)$