 The map

formula_35

is a homomorphism of rings.An integral notation is often used for formula_36, as in

The theorem is also correct for unbounded measurable functions "f", but then formula_36 will be an unbounded linear operator on the Hilbert space "H".The spectral theorem says that every self-adjoint operator formula_39 has an associated projection-valued measure formula_40 defined on the real axis, such that
This allows to define the Borel functional calculus for such operators