Positivity for Hermitian dot product?

Question

One of the requirements for the Hermitian dot product is positivity, i.e. $||v||^2 \ge 0, $ and $||v||^2 = 0 \iff v=0$. I was wondering what exactly this means in the complex numbers. Does it mean just the real part of the dot product must be positive, or that the imaginary part must also be 0?

Answer 1

The positivity is about $h(x,x)$. In general, $h(x,y)$ is a complex number so it doesn't make sense to talk about positivity, but $h(x,x)$ is always a real number because of $h(y,x) = \overline{h(x,y)}$, so it makes sense to ask that it must be positive.