Riesz–Fischer theorem

The Riesz–Fischer theorem is the structural cornerstone of L^p theory: it certifies that the L^p spaces have no ‘holes’. A sequence of functions whose successive differences shrink to zero in L^p norm — a Cauchy sequence — is guaranteed to home in on an actual limit function that itself lives in L^p. In short, you can take limits inside L^p without falling out of it.

Stated for 1 <= p <= infinity: the space L^p(X, mu) is complete, so every Cauchy sequence in the L^p norm converges in that norm to an element of L^p. Equivalently, L^p is a Banach space. The classical proof for finite p extracts a rapidly converging subsequence, builds a candidate limit by summing the absolute differences (controlled via the monotone convergence theorem so the sum is finite almost everywhere), and shows the whole sequence converges to it.

Historically the name attaches to the Fourier-analytic incarnation: given any square-summable sequence of would-be Fourier coefficients, there exists an L^2 function having exactly those coefficients against a fixed orthonormal system. This made L^2 the correct home for Fourier series and was a decisive early triumph of the Lebesgue integral over the Riemann integral, whose function spaces are not complete in these norms.