= Continuity and uniqueness of holomorphic functional calculus
{title2=$\Theta_x:\mathcal O(U)\to A$}
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 $\Gamma$ gives
$$
\|\Theta_x(f)\|\leq\frac{\operatorname{length}(\Gamma)}{2\pi}\max_{z\in\Gamma}\|(z1-x)^{-1}\|\max_{z\in\Gamma}|f(z)|,
$$
so the <holomorphic functional calculus> is <continuous>.
It is the unique \b[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.
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