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 Lp 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 ψn is a Cauchy sequence in Lp, then there exists an essentially unique (i.e. unique except possibly on a set of measure zero) function ψ in Lp such that ψn 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 ck, 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 L2, then its Fourier transform is also in L2 and that the norms of a function and its Fourier transform are equal.