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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
1
vote
Spectral theorem for unbounded self-adjoint operators on REAL Hilbert spaces
I posted a solution at StackExchange that uses only real techniques, but I don't know of any reference that uses the technique.
3
votes
Who first used the multiplication operator version of spectral theory
Dieudonne records the following in Chapter VII of History of Functional Analysis:
Following the method of von Neumann, it is then easy to extend the homomorphism $f \mapsto f(N)$ to the algebra $\ …