I am reading the book "A Hilbert Spaces Problem book", written by P. Halmos.
In chapter 4, the author define Hardy space $H^2$ as the following.
Let $C$ be the unit circle in the complex plane $C= \{ z : |z|=1 \}$ and let $\mu$ be Lebesgue measure on the Borel sets of $C$, normalized so that $\mu(C)=1$.
If $e_n(z)=z^n$ for $|z|=1$ ($n=0, \pm 1, \pm 2, \dots)$, then the functions $e_n$ form an orthonormal basis for $L^2$.
Question: I tried to prove that it forms an orthonormal basis but it is impossible. Could you help me please?
The space $H^2$ is the subspace of $L^2$ spanned by the $e_n$'s with $n \ge 0$.
Question: I heard that each element in $H^2$ is not only a function but also a represetative of some class. However, I do not know which equivalent relation is used in here. Could you help me please?