Basis for $L^2(0,1)$ and $H^2(0,1)\times L^2(0,1)$ defined on complex field

Question

We know $e_n=\sin(n\pi x)$ forms an orthogonal basis for the real Hilbert space $L^2(0,1)$. I am wondering how to modify this to be a basis for complex space $L^2(0,1)$.

Also, is $e_n=(\frac{1}{n^2\pi^2} \sin(n\pi x),sin(n\pi x))$ a basis for $H^2(0,1)\times L^2(0,1)$ defined on a real field. I suspect $e_{-n}=(\frac{-1}{n^2\pi^2} \sin(n\pi x),sin(n\pi x))$ is linearly independent from all of $e_n$'s, so it should be included. What would the basis on a complex field?

Answer 1

$\{e^{2 \pi inx}:n \in \mathbb Z\}$ is a well-known orthonormal basis for complex $L^{2}(0,1)$. Any book on Fourier series has a proof. Another orthonormal basis you can derive using this is $\{\sin (nx):n \in \mathbb N\}\cup \{\cos (nx):n \in \{0\} \cup \mathbb N\}$.