Orthonormal basis for $L^2[a,b]$

Question

Is there an orthonormal basis for $L^2[a,b]$, as $ \left\{e^{2 \pi i n x}\right\}_{n=-\infty}^{\infty} \text { is a complete orthonormal basis } $ in $L^2[0,1]$?

Answer 1

Yes, use the map from $[0, 1]$ to $[a, b]$.

Answer 2

$e^{2\pi in (\frac {x-a} {b-a})}, n \in \mathbb Z$ is one.

This is for normalized Lebesgue measure. If you are using Lebesgue measure on $[a,b]$ you have to divide these functions by $\sqrt {b-a}$ to make their norms equal to $1$.