Is a Complex Inner Product defined to be anti-linear (conjugate linear) to both arguments corrrect?

Question

I know that we can have complex inner products defined such as

• Conjugate linear to first argument: SUM(conj(lhsElement), rhsElement)
• Conjugate linear to second argument: SUM(lhsElement, conj(rhsElement))

Can we have a complex inner product defined to be conjugate linear to both arguments, such as SUM(conj(lhsElement), conj(rhsElement))?

Answer 1

No, you can't, otherwise your inner product won't be positive definite. Indeed, let $\langle\,\cdot\,,\cdot\,\rangle : V \times V \rightarrow \mathbb{C}$ be a hermitian "bi-anti-linear" form. Then, taking a vector $v \in V$ such that $\langle v,v \rangle > 0$, then $\langle iv,iv \rangle = -\langle v,v \rangle < 0$, and vice versa.