Continuity and uniqueness of holomorphic functional calculus (${\Theta }_x:\mathcal{O}(U)\to A$)

Equip the space $\mathcal{O}(U)$ of holomorphic functions on an open neighborhood $U$ of ${\sigma }_A(x)$ with the compact-open topology. A fixed admissible contour $\Gamma$ givesso the holomorphic functional calculus is continuous.

It is the unique continuous unital algebra homomorphism from $\mathcal{O}(U)$ to $A$ sending the coordinate function to $x$. A unital algebra homomorphism must send each inverse coordinate difference to the corresponding resolvent of an element, so its values on rational functions with poles outside $U$ are forced. The Runge theorem makes those rational functions dense for the compact-open topology; continuity then forces all remaining values. This last step requires the stated continuity hypothesis.