I was reading the following text about the mathematical foundations of quantum mechanics when I stumbled upon the following conditions that the inner product must satisfy:
I woud like to understand why the above conditions must be satisfied, and resolve some seeming conflicts.
I always associated the inner product in quantum mechanics with an overlap integral, this notion would intuitively imply that it should be symmetric in exchange of $\psi$ and $\omega$ because the overlap of the two functions should not be a function of how I label them. Why do I require the complex part of the inner product to be anti-symmetric with respect to swapping of the arguments?
I understand that choosing it to be linear in its second argument is just a matter of convention and it makes the Dirac notation work nicer, as operators can then act on the elements to the right.
Again, this is not clear to me why I should require the inner product to be antilinear in its first argument. Is it just because it is a necessary requirement to satisfy the condition 4 below?
This makes sense.
I would be very glad if someone could explain the reason for the first condition to me, as it seems mystifying to me why it is necessary to impose it.