People seem to really like replacing sums by integrals. It turns out that even something like a direct sum of Hilbert spaces can be generalized. The decomposition of the regular representation of the circle (which is a Lie group) into irreducible unitary representations gives us the theory of Fourier series. One thing that confused me for a while is another thing called the “Fourier transform”, which is when you write a function on (rather than a function on the circle/periodic function on ) as an *integral* rather than a series. However, I realized it’s completely natural to do so: the analogous decomposition for the additive group of real numbers (which is also a Lie group) gives something a bit different: you can’t decompose as an orthogonal direct sum like you did to , but you *can* decompose it into something called a “direct integral”…

Suppose you have a set (which will act as an “index set” for your spaces) equipped with a non-negative measure, , and for each you have an associated Hilbert space . Suppose each is of the same dimension, so we can identify them all with one space . We denote by the space of functions such that for any , the function is -measurable, and the function is square-integrable w.r.t. , that is,

Then becomes (as one can show) a Hilbert space under the obvious pointwise operations of vector addition and scalar multiplication, and the scalar product

is called the **continuous direct sum** (or **direct integral**) of the spaces with respect to , and we write it as

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## About mlbaker

just another guy trying to make the diagrams commute.

And that is where the fun begins: F.e. every unitary representation of a locally compact group on a Hilbert space is a direct integral of irreducible representations. The simplest example is probably the fact that by using the Fourier transform one has that the representation of on is nothing else but the direct integral of copies of , or more precisely of the Hilbert spaces on which clearly acts.