Having problem understanding the complex fourier series

Question

Here is my textbook, and I am wondering where that conjugate of $g(x)$ comes from? Why it isn't the integral of $f(x)g(x)$?

It seems to contradict this theorem, I am not quite sure

Answer 1

Why would it be the integral of $f(x)g(x)$? The equation $$\langle f,g\rangle=\int_{-L}^Lf(x)\bar g(x)dx$$ is a definition of the inner product. The reason is because one of the inner product axioms requires that $\langle f,g\rangle=\overline{\langle g,f\rangle}$ (what Wikipedia calls conjugate symmetry), which is not satisfied if we define the inner product to be the integral of $f(x)g(x)$.