Definition of inner product space

Question

In the definition, we defined linearity in the first argument, Hermitian symmetry. And these two imply anti-linearity in second argument. Is it equivalent, if I cancel the Hermitian symmetry and only define linearity and anti-linearity in first and second argument respectively? Namely,

$(x,ay+bz)=a^*(x,y)+b^*(x,z)\wedge(ax+by,z)=a(x,z)+b(y,z)$

$\Rightarrow (y,x)=(x,y)^*$

Answer 1

No it's not.

For instance, on $\mathbb{C}^n$, take any matrix $M\in M_n(\mathbb{C})$ that is not hermitian, and put $(X,Y)\mapsto X^t M \overline{Y}$.