This question was posed on MathStackExchange but did not get an answer (even with a bounty).
In all books that I have checked the spectral theorem (every self-adjoint unbounded operator on a Hilbert space is unitary equivalent to a multiplication operator on some $L_2(\mu)$) is only stated for complex Hilbert spaces (and the use of the Cayley transformation for the reduction to the bounded case requires indeed complex scalars). However, since the spectrum of a a self-adjoint operator is real the theorem should (or could) be true in real Hilbert spaces. I could imagine an argument by complexification but there are a number of things to do, in particucar, I think that the unitary operator between the complexified Hilbert space and the complex $L_2(\mu)$ does not automatically map the original real Hilbert space to the real-valued functions). Hence the question:
Is the the spectral theorem in real Hilbert spaces true? (In this case I would also like to have a reference.)
Best Answer
Simon Henry's comment is close to an answer. As I asked for a reference :
Remark 20.18 in R. Meise and D. Vogt, Introduction to Functional Analysis.