Theorem 2 (E. Bishop)

— *Let $X$ be a compact space, $\mathscr{A}$ a closed subalgebra of the complex Banach algebra $\mathscr{C}(X; \mathbf{C})$. Assume that $\mathscr{A}$ contains the constants and separates the points of $X$. Let $a$ be a point of $X$; the following conditions are equivalent:

a) There exists a function $f \in \mathscr{A}$ such that $|f(x)| < |f(a)|$ for all $x \neq a$.

b) The point $a$ is $\mathscr{A}$-extremal and admits a countable fundamental system of neighborhoods.

a) $\Rightarrow$ b): Let $f \in \mathscr{A}$ be such that $|f(a)| > |f(x)|$ for $x \neq a$; by Prop. 9 of No. 4, $a$ is an $\mathscr{A}$-extremal point. On the other hand, if $U_n$ is the set of $x \in X$ such that $|f(x)| > |f(a)| - 1/n$, then $U_n$ is an open neighborhood of $a$, and the intersection of the $U_n$ reduces to $a$; since $X$ is compact, the $U_n$ form a fundamental system of neighborhoods of $a$ (GT, I, §9, No. 1, Th. 1).

b) $\Rightarrow$ a): It suffices to verify that b) implies the hypotheses of Lemma 4. With the notations of that lemma, set $\varepsilon = \log d / \log c$; thus $0 < \varepsilon < 1$. Since $a$ is an $\mathscr{A}_r$-extremal point, there exists a function $g \in \mathscr{A}$ such that

(No. 3, Prop. 6, b)). Set $f = c^g$; since $f$ is the sum of the normally convergent series $\sum_{n=0}^{\infty} (\log c)^n g^n / n!$, we have $f \in \mathscr{A}$ and

Q.E.D.