I am trying to work out the difference between hermitian and conjugate linear sesquilinear form. Let me elaborate on my confusion:
Let $H$ be a Hilbert space. One definition (see e.g. here page 49) for sesquilinear form is that it is a map $H^2 \to \mathbb C$ such that it is linear in the first argument and conjugate linear in the second.
My question is why is this not equivalent to this?:
A sesquilinear form is a map $H^2 \to \mathbb C$ such that it is linear in the first argument and such that for all $x,y\in H$ we have $\langle x,y\rangle = \overline{\langle y,x \rangle}$?
A sesquilinear form with the property $\langle x,y\rangle = \overline{\langle y,x \rangle}$ is called hermitian. Since we have extra terminology it would seem that one would define a sesquilinear map as a hermitian map that is linear in the first argument. But I looked and every definition I could find described it as a map "linear in the first and almost linear in the second argument". Which strongly suggests that my own idea for a definition is not equivalent.
If $(x,y)$ is the inner product then $s(x,y)=i(x,y)$ is linear in the first coordinate and conjugate linear in the second. However $\overline{s(x,y)}\ne s(y,x)$.