**Notation Used By Mathematicians**

The object physicists are considering when using the "bra-ket" notation is a Hilbert space (a complete inner product space).

Let be a Hilbert space and is a vector in . What physicists would denote as is the vector itself. That is

- .

Let be the dual space of . This is the space of linear functionals on . The isomorphism is defined by where for all we have

- ,

where and are just different notations for expressing an inner product between two elements in a Hilbert space (or for the first three, in *any* inner product space). Notational confusion arises when identifying and with and respectively. This is because of literal symbolic substitutions. Let and let . This gives

One ignores the parentheses and removes the double bars. Some properties of this notation are convenient since we are dealing with linear operators and composition acts like a ring multiplication.

Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they don't use the *-symbol, but an overline (which the physicists reserve to averages) to denote conjugate-complex numbers, i.e. for scalar products mathematicians usually write

whereas physicists would write for the same quantity

Read more about this topic: Bra-ket Notation