Editing Borel functional calculus (section)

== The general functional calculus ==
{{see also|Operational calculus}}
We can also define the functional calculus for not necessarily bounded Borel functions ''h''; the result is an operator which in general fails to be bounded.  Using the multiplication by a function ''f'' model of a self-adjoint operator given by the spectral theorem, this is multiplication by the composition of ''h'' with ''f''.

{{math theorem | Let ''T'' be a self-adjoint operator on ''H'', ''h'' a real-valued Borel function on '''R'''. There is a unique operator ''S'' such that
<math display="block">\operatorname{dom} S = \left\{\xi \in H: h \in L^2_{\nu_\xi}(\mathbb{R}) \right\}</math>
<math display="block">\langle S \xi, \xi \rangle = \int_{\mathbb{R}} h(t) \ d\nu_{\xi} (t), \quad \text{for} \quad \xi \in \operatorname{dom} S</math>}}

The operator ''S'' of the previous theorem is denoted ''h''(''T'').

More generally, a Borel functional calculus also exists for (bounded) normal operators.