Let \(n\in \mathbb{N}\). Let \(E_n\) be the set of all roots of unity for \(n\). Consider the map \(k\mapsto \e^{2\pi i\frac{k}{n}}. \) The vector space \(V=\{f\colon E_n \to \mathbb{C}\}\) endowed with the scalar product
\[ \langle f, g\rangle=\sum_{k\in \mathbb{Z}_n} f(k)g(k) \]is a Hilbert space .
Remarks
- The vectors \(e_l=(e_l(1),\ldots ,e_l(n))\), where \(e_l\) defined for the Fourier matrix , are orthogonal, i.e. \[ \langle e_l, e_k\rangle=n\delta_{lk}. \]
- The vectors \(e_l^*=\frac{1}{\sqrt{n}}e_l\) form an orthonormal basis.