I'll have to make a few assumptions in order to answer this question.

1. I assume that the vectors are supposed to form an orthonormal basis for the Hilbert space.

2. I assume that you are using the physicists' convention that the inner product is linear in the second variable and conjugate-linear in the first variable (mathematicians always use the opposite convention: linear on the left and conjugate-linear on the right).

Under those assumptions,