Generalized Fourier Coefficients and Fourier Series in Hilbert Spaces
Definition[^1]
Let $H$ be a Hilbert space, and let $\left\{ u_{\alpha} \right\}_{\alpha\in A}$ be a normal orthogonal system in $H$. Then, for a fixed $x\in H$, let’s define the complex function $\hat{x} :A\to \mathbb{C}$ as follows.
$$ \hat{x}(\alpha)=\left\langle x,u_{\alpha} \right\rangle $$
The values above are referred to as the Fourier coefficients of $x$ with respect to $\left\{ u_{\alpha} \right\}$.