One form of the spectral theorem concerns a complex Hilbert space 
. Let 
denote the set of bounded
linear operators from 
to itself. An operator 
is normal if 
, where 
is its adjoint operator.
For such 
,
there is a unique resolution of the identity 
on the Borel subsets of its operator spectrum 
satisfying
 |
Here a resolution of the identity is an orthogonal projection-valued measure with 
, where 
is the identity operator.
For every vector 
, 
is countably
additive in the norm of 
(Rudin 1991, pp. 316-318). Furthermore, every orthogonal
projection 
commutes with every 
that commutes with
(Rudin 1991, pp. 324-325).
