For a given [[Fourier series]] we define an [orthonormal bases](Orthonormal%20bases.md) that is the set of exponential terms $e^{ikx}$, where $k=\frac{2\pi n}{L}$ and $n\in \mathbb{Z}$. The elements of this set are called _fourier modes_. # Orthonormality of Fourier modes # Completeness relation The sum of the Fourier modes in a given Fourier series form the completeness relation: $\delta(x)=\frac{1}{L}\sum_k e^{ikx}$ where $\delta(x)$ is the [[dirac delta function]]. --- # Proofs and examples ## Proof of the completeness relation #MathematicalFoundations/Analysis/FourierAnalysis