 In the language of Hilbert spaces, the set of functions formula_145 is an orthonormal basis for the space formula_146 of square-integrable functions on formula_147.This space is actually a Hilbert space with an inner product given for any two elements formula_49 and formula_101 by
The basic Fourier series result for Hilbert spaces can be written as

This corresponds exactly to the complex exponential formulation given above.The version with sines and cosines is also justified with the Hilbert space interpretation.