We continue discussing Dirac notation by bringing up the inner product. To this point, our notation applies to any vector space and its dual, with the ket denoting a vector and the bra denoting a linear functional . The evaluation is then denoted by the bra-ket pairing .
But the neat thing about this notation is that it makes bras look like some sort of reflection of kets. And they are, in a sense. The dual space is some sort of reflection of the vector space , but there’s no clear mapping from vectors in one space to vectors in the other; unless, that is, we pick a specific isomorphism; or, equivalently, an inner product.
When we’ve got an inner product in the picture, we get a (conjugate) linear isomorphism that sends the vector to the linear functional . In Dirac notation, we send the ket to the bra . Then the value of this linear functional on a vector (the ket ) is the pairing , just as it should be.