Adjoint definition and inner product

Question

Can someone explain the last equality?

Isn't it $\overline{(Te_j, e_i)} = (e_i, Te_j)$? How is this the definition of adjoint

Answer 1

Yes, there is a typo in the quoted formula. The second equality should indeed read $\overline{(Te_j, e_i)} = (e_i, Te_j)$, which is just the symmetry property of the complex inner product.

The first equality should read $(T^*e_i, e_j)=\overline{(Te_j, e_i)}$, which follows from the definition of matrix elements and the elementwise definition of adjoint: $(T^*)_{ij}=\overline{(T)_{ji}}$.

Correcting these two mistakes makes it reasonable. Of course something else may have been intended; I can only guess from that amount of context. Looking at the source provided by the OP, it seems that there are indeed two typos in the same formula.

The adjoint $T^*$ of $T$ is the one and only linear operator which satisfies $(T^*e_i, e_j)=(e_i, Te_j)$. This is the "modern day definition of adjoint" meant in the text. More typically, the adjoint is defined so that $(T^*a, b)=(a, Tb)$ for all vectors $a$ and $b$, but it is enough to demand that it holds for the basis.