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|
The converse of Parseval's theorem is also true: if ck
are any numbers such that ∑ |ck
< ∞, then there is an essentially unique function in L2
that has ck
as its Fourier coefficients. Essentially this states that the l2
space of infinite sequence
s of complex numbers whose absolute square sum is convergent
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.