Quantum Mechanics, The Theoretical Minimum

I am delighted so far with Susskind & Friedmanā€˜s book. It actually has equations! I have one quibble; as they introduce the notion of dual space they leave the impression that there is a natural map between a space and its dual. The two spaces are isomorphic but there is no non-trivial automorphism as there is for complex numbers!
Developement here

A footnote on page 33 remarks that in math, basis vectors are not required to be orthonormal while in physics they are. The difference, I think, is that with a quadratic form it is possible and desirable to require orthonormal. They smuggled in a quadratic form as they introduced their dual concept. Raw vectors have no length!