If you ever wondered about physicists’ convention of defining sesquilinear forms (e.g. inner products on a complex space) to be linear in the second variable and conjugate linear in the first, the reason they do it is because of Dirac notation: they write the result of applying the covector (linear functional) to the vector as the inner product . Contrast this to the way mathematicians usually assign and declare the inner product to be conjugate linear in the second variable. Even as a pure mathematician, the former now seems more natural to me, especially considering how commonly vector spaces are paired with their duals, and the fact that function application is written on the left e.g. as (not ; sorry Victor, enjoy your ). To be exact, what I’m saying is that this function-application-on-the-left convention makes it more natural for bras to represent functionals and kets to represent vectors, and not vice versa.
Also, the Borel -algebra is not in any way related to the real numbers; you can talk about it over any topological space: just take the -algebra generated by the compact sets (in almost all decent cases, this is equivalent to the -algebra generated by the open sets).