Is basis of Laurent functions orthonormal in any sense( for example $L^2([-R,R])$? For complex Fourier series applies following: "In the language of Hilbert spaces, the set of functions {$e_n = e^{inx}$; $n \in \mathbb{Z}$} is an orthonormal basis for the space $L^2([−\pi, \pi]$) of square-integrable functions of [−$\pi$, $\pi$]. " - Hilbert space interpretation
But for power series( for example Taulor l. Laurent series) " the set of functions " that form Laurent series are what? In other words is basis of Laurent functions orthonormal in any sense( for example $L^2([-R,R])$?
 A: *

*Since $x^2$ and $x^4$ are both positive on $[-R,R]$, they cannot be orthogonal in any weighted Lebesgue space on real line, i.e., a space with inner product $\langle f,g\rangle = \int_a^b fgw$ where $w$ is a weight on $[a,b]$.  

*In the complex plane the monomials $z^k$ are orthonormal with respect to the inner product $\langle f,g\rangle =\frac1{2\pi}\int_{|z|=1}f\bar g$, which is structure of the Hardy space $H^2$. However, a closer look will reveal that this is the same Fourier series stuff in different notation. 

*We can try to stay within real numbers and define inner product of polynomials differently, using the vector of coefficients. That is, write polynomials as $p(x)=\sum_{k=0}^\infty a_k x^k$  and $q(x)=\sum_{k=0}^{\infty} b_k x_k$, where only finitely many coefficients are nonzero. Define $\langle p,q\rangle =\sum_{k=0}^\infty a_kb_k$. This is an inner product with respect to which the monomials $x^k$ form an orthonormal basis. The completion of this space is a subspace of the Hardy space $H^2$ which consists of the functions that are real on the real line. In other words, we are led to the complex plane whether we want it or not.
