Abstract Fourier series and Parseval’s identity
1 Orthonormal basis
Let be a separable complex Hilbert space.11 1 One talk do everything we are doing and obtain the same results for nonseparable Hilbert spaces, but one has to define what uncountable sums mean. This is done in John B. Conway, A Course in Functional Analysis, second ed., chapter I. If , , and , we say that the set is orthonormal. If is a dense subspace of , we say that is an orthonormal basis for . We can write this in another way. If are subsets of , let be the closure of the span of . To say that is an orthonormal basis for is to say that is orthonormal and that .
2 Abstract Fourier series
If and the sequence converges in , we denote its limit by
This is a definition of an infinite sum in . Since is complete, one usually shows that a sequence converges by showing that the sequence is Cauchy, and hence to show that converges it is equivalent to show that
as . And showing this is equivalent to showing that
as . This is equivalent to
as , and this is equivalent to the series
converging. Thus, the series converges if and only if the series converges.22 2 Furthermore, using the triangle inequality rather than the orthonormality of the , one can check that if the series converges then the series converges.
Let be an orthonormal basis for ; it is a fact that one exists. Let and define
It follows that
where we used in the third line. Therefore the series converges, and so the sequence converges to some . Since converges to , in particular it converges weakly to , i.e., for any ,
Therefore for any ,
this is because for we have and hence . As for all , it follows that , i.e. . Hence,
We call this an abstract Fourier series for .33 3 If , one checks that , is an orthonormal basis for . Then, and is the limit in of . Thus in , It can be written as
and thus can be written without as
is a projection with rank 1, and the above series conveges in the strong operator topology on . Writing the identity map in this way is called a resolution of the identity.
3 Parseval’s identity
On the one hand
On the other hand,
which is Parseval’s identity.