Dirac’s Dual Space

Dirac’s dual concept, at least as expressed here, might be slightly at odds with that of Halmos. Lets take the space V which is the set of pairs of rational numbers: Q². The field is Q. The only ‘obvious basis’ for V is {(1, 0), (0, 1)}. V’s dual, V*, according to Halmos, is the set of linear maps from V to Q:
{λ(x, y).(ax+by) ⎥ a∊Q ⋀ b∊Q}. In the Halmos scheme we can have a vector space and its dual with no quadratic form in sight. The quadratic form is smuggled in along with the dual in the Dirac scheme.

In the Halmos scheme any linear bijection between B a vector space and its dual establishes a quadratic form Q. For x and y in V, Q(x, y) = B(y)(x). (B(y) is in V* and is thus a function with V as its domain.)

There is something else significant: the Dirac vision exploits an automorphism in the complex numbers. The book introduces the notation “⟨A|z” without explanation. Does the “A” within “⟨A|” denote something as “2” within “2+4” denotes two? Which one is the vector? What does it mean to write a vector, ⟨A|, to the left of a field element z? The equation “⟨A|z = z*⟨A|” seems to me to be a definition of notation rather than an axiom. In general the fields of many other vector spaces have either no automorphisms, or many.

I think that the book does not follow thru on the promise to distinguish between a vector space and its dual, or as they would say a bra and a ket vector. That’s OK for there is a natural 1-1 bijection between them.