Editing Borel functional calculus (section)

=== Existence of a functional calculus ===
The existence of a mapping with the properties of a functional calculus requires proof. For the case of a bounded self-adjoint operator ''T'', the existence of a Borel functional calculus can be shown in an elementary way as follows:

First pass from polynomial to [[continuous functional calculus]] by using the [[Stone–Weierstrass theorem]]. The crucial fact here is that, for a bounded self adjoint operator ''T'' and a polynomial ''p'',
<math display="block">\| p(T) \| = \sup_{\lambda \in \sigma(T)} |p(\lambda)|.</math>

Consequently, the mapping
<math display="block"> p \mapsto p(T) </math>
is an isometry and a densely defined homomorphism on the ring of polynomial functions. Extending by continuity defines ''f''(''T'') for a continuous function ''f'' on the spectrum of ''T''. The [[Riesz-Markov theorem]] then allows us to pass from integration on continuous functions to [[spectral measure]]s, and this is the Borel functional calculus.

Alternatively, the continuous calculus can be obtained via the [[Gelfand transform]], in the context of commutative Banach algebras. Extending to measurable functions is achieved by applying Riesz-Markov, as above. In this formulation, ''T'' can be a [[normal operator]].

Given an operator ''T'', the range of the continuous functional calculus ''h'' → ''h''(''T'') is the (abelian) C*-algebra ''C''(''T'') generated by ''T''. The Borel functional calculus has a larger range,  that is the closure of ''C''(''T'') in the [[weak operator topology]], a (still abelian) [[von Neumann algebra]].