Editing Borel functional calculus (section)

== Resolution of the identity ==
Let <math>T</math> be a self-adjoint operator. If <math>E</math> is a Borel subset of '''R''', and <math> \mathbf{1}_E </math> is the [[indicator function]] of ''E'', then <math> \mathbf{1}_E(T) </math> is a self-adjoint projection on ''H''.  Then mapping
<math display="block"> \Omega_T: E \mapsto \mathbf{1}_E(T)</math>
is a [[projection-valued measure]]. The measure of '''R''' with respect to <math display="inline">\Omega_T</math> is the identity operator on ''H''. In other words, the identity operator can be expressed as the spectral integral
:<math>I = \Omega_T([-\infty,\infty]) = \int_{-\infty}^{\infty} d\Omega_T</math>.

Stone's formula<ref>{{cite journal |last1=Takhtajan |first1=Leon A. |title=Etudes of the resolvent |journal=Russian Mathematical Surveys |date=2020 |volume=75 |issue=1 |pages=147–186 |doi=10.1070/RM9917 |url=https://arxiv.org/abs/2004.11950v1|arxiv=2004.11950 }}</ref> expresses the spectral measure <math>\Omega_T</math> in terms of the [[Resolvent formalism|resolvent]] <math>R_T(\lambda) \equiv \left(T-\lambda I \right)^{-1}</math>:
:<math>\frac{1}{2\pi i} \lim_{\epsilon \to 0^+} \int_a^b \left[ R_T(\lambda+i\epsilon)) - R_T(\lambda-i\epsilon) \right] \, d\lambda = \Omega_T((a,b)) + \frac{1}{2}\left[ \Omega_T(\{a\}) + \Omega_T(\{b\}) \right].</math>

Depending on the source, the '''resolution of the identity''' is defined, either as a projection-valued measure <math>\Omega_T</math>,<ref>{{cite book | last=Rudin | first=Walter | title=Functional Analysis | publisher=McGraw-Hill Science, Engineering & Mathematics | publication-place=Boston, Mass. | date=1991 | isbn=978-0-07-054236-5|pages=316–317}}</ref> or as a one-parameter family of projection-valued measures <math>\{\Sigma_\lambda\}</math> with <math>-\infty < \lambda < \infty</math>.<ref>{{cite book | last=Akhiezer | first=Naum Ilʹich | title=Theory of Linear Operators in Hilbert Space | publisher=Pitman | publication-place=Boston | date=1981 | isbn=0-273-08496-8|page=213}}</ref>

In the case of a discrete measure (in particular, when ''H'' is finite-dimensional), <math display="inline">I = \int 1\,d\Omega_T</math> can be written as
<math display="block">I = \sum_{i} \left | i \right \rangle  \left \langle i \right |</math>
in the Dirac notation, where each <math>|i\rangle</math> is a normalized eigenvector of ''T''. The set <math> \{ |i\rangle \}</math> is an orthonormal basis of ''H''.

In physics literature, using the above as heuristic, one passes to the case when the spectral measure is no longer discrete and write the resolution of identity as
<math display="block">I = \int\!\!   di~ |i\rangle \langle i|</math>
and speak of a "continuous basis", or "continuum of basis states", <math> \{ |i\rangle \}</math> Mathematically, unless rigorous justifications are given, this expression is purely formal.