= Solution
For nonnegative integers $m,n$, <Fourier orthogonality> gives
$$
\langle z^m,z^n\rangle=\frac1{2\pi}\int_0^{2\pi}e^{i(m-n)\theta}\,d\theta=\delta_{mn}.
$$
Every <polynomial> is a finite linear combination of these <monomials>, and distinct <monomials> are <linearly independent>. Therefore
$$
\boxed{1,z,z^2,z^3,\ldots\quad\text{is an orthonormal basis of }P.}
$$
This is an algebraic <basis>: no infinite linear combinations are needed in the <polynomial> <vector space>.
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