Riesz-Fischer Theorem

The Riesz-Fischer Theorem states that the space of square-integrable functions, and more generally the space of square-summable sequences of coefficients, is complete, meaning every Cauchy sequence of such functions converges to a limit that is again square-integrable. Named for Frigyes Riesz and Ernst Sigismund Fischer, who proved it independently in 1907, it established that the space of square-integrable functions is a genuine Hilbert space, a fact essential to the mathematical foundations of Fourier analysis and quantum mechanics.