Supremum bound for a complete function-algebra norm ($\Vert f{\Vert }_{\infty }\le \Vert f\Vert$)

For an algebra of complex-valued functions with a complete algebra norm, every point value $f(t)$ belongs to the spectrum of an element of $f$ in the artificial unitization. Indeed $(h,\lambda )\mapsto h(t)+\lambda$ is an algebraic unital character of an algebra, and it vanishes on $f-f(t)1$. The spectrum bound gives $|f(t)|\le \Vert f\Vert$ without first assuming that evaluation is continuous.