An important result, which makes a lot of the extra complexity in the definition of the Lebesgue integral all worth it. It basically states that the L_{p} spaces, i.e. the spaces of all pth-power functions which are integrable, form a Banach space given the norm ||ψ|| = ∫ (|ψ(x)|^{p})^{1/p} dx. In other words, if a sequence of functions ψ_{n} is a Cauchy sequence in L_{p}, then there exists an essentially unique (i.e. unique except possibly on a set of measure zero) function ψ in L_{p} such that ψ_{n} converges to ψ in the mean.

An important consequence of this theorem is a very useful result in the theory of Fourier series and Fourier transforms, which is also sometimes called the Riesz-Fischer Theorem, part of which is more commonly known as Parseval's Theorem. The theorem states that if the complex Fourier coefficients of a function are c_{k}, then the following holds:

∞ 2 2
∑ |c | = ∫ |f|
k=-∞ k

The converse of Parseval's theorem is also true: if c

_{k} are any numbers such that ∑ |c

_{k}|

^{2} < ∞, then there is an essentially unique function in L

_{2} that has c

_{k} as its Fourier coefficients. Essentially this states that the l

^{2} space of

infinite sequences of complex numbers whose absolute square sum is

convergent is

isomorphic to L

_{2}.

This theorem formed the basis for the proof of the equivalence of the Schrödinger and Heisenberg Pictures in Quantum Mechanics.

An analogous result for Fourier transforms states that if ψ is in L_{2}, then its Fourier transform is also in L_{2} and that the norms of a function and its Fourier transform are equal.