$\left\{ e _ { i } \right\}$ is an orthonormal basis which has the orthonormal condition as following: $$e _ { i } ^ { T } \cdot e _ { j } = \delta _ { i j }$$ In Dirac Notation where $| i \rangle = | e _ { i } \rangle$ the condition can be written as $$\langle i | j \rangle = \left( e _ { i } ^ { T } \right) ^ { * } \cdot e _ { j } = \delta _ { i j }$$

Does that mean $\left( e _ { i } ^ { T } \right) ^ { * } = e _ { i } ^ { T }$? which could only be true when $\left\{ e _ { i } \right\}$ are made up of real vectors, isnt it?