TITLE: Spectral theorem for unbounded self-adjoint operators on REAL Hilbert spaces
QUESTION [14 upvotes]: In all books that I have checked the spectral theorem (every self-adjoint 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 be true in real Hilbert spaces. I can imagine an argument by complexification but there a number of things to do). Hence the question:
Is there an easily accessible reference to the spectral theorem in real Hilbert spaces?

The question is now also discussed on MathOverflow.

REPLY [2 votes]: Please consult
R. K. GOODRICH, The spectral theorem for real Hilbert space, Acta Sci. Math.
(Szeged), 33 (1972), 123–127.
(open access)
Miroslav