It is L^2 for infinite dimensional Hilbert spaces like position or momentum, where quantum states become functions.
For finite dimensional Hilbert spaces such as spins, quantum state is a complex vector rather than a function, and the "braket" is not an integral but vector product.
But I don't think that's even the point. The whole article is a rationalization of basic rules of quantum mechanics; when I look back at the history of physics, rationalization of known --at the time-- physical laws (which are often replaced by "better laws" after a while when get a better we understanding of nature) is most often counter-productive.
But I don't think that's even the point. The whole article is a rationalization of basic rules of quantum mechanics; when I look back at the history of physics, rationalization of known --at the time-- physical laws (which are often replaced by "better laws" after a while when get a better we understanding of nature) is most often counter-productive.