Lemma 4

Let X be a compact space, $\mathscr{H}$ a closed linear subspace of the Banach space $\mathscr{C}(X; \mathbf{C})$ (resp. $\mathscr{C}(X; \mathbf{R})$). Let a be a point of X admitting a countable fundamental system of neighborhoods; assume that, for any numbers c and d such that $0 < c < d < 1$ and any open neighborhood U of a, there exists an $f \in \mathscr{H}$ such that

Then there exists a function $u \in \mathscr{H}$ such that $|u(x)| < |u(a)|$ for all $x \neq a$.

Let $(V_n)$ ($n \geq 1$) be a fundamental system of neighborhoods of $a$, and let $\lambda, \mu, \varepsilon$ be numbers such that

Thus $0 < \lambda / \mu < 1 / \mu < 1$. We are going to define, by induction on $n$ ($n \geq 1$), a decreasing sequence $(U_n)$ of open neighborhoods of $a$ such that $U_n \subset V_n$ for all $n$, and a sequence $(h_n)$ of functions in $\mathscr{H}$ satisfying the relations Assume $h_m$ and $U_m$ defined for $1 \leq m < n$, satisfying the four preceding conditions (with $n$ replaced by $m$); on the other hand, set $U_0 = X$. The function $\sum_{j=1}^{n-1} \lambda^j h_j$ (equal to 0 if $n = 1$) is continuous and takes the value $\sum_{j=1}^{n-1} \lambda^j$ at the point $a$; therefore there exists an open neighborhood $U_n$ of $a$, contained in $U_{n-1} \cap V_n$, such that By hypothesis there exists a function $f \in \mathscr{H}$ such that Set $h_n = f / f(a)$; the relations $(13_n)$, $(14_n)$ and $(15_n)$ then hold; set By (17) and $(13_n)$, for $y \in U_n$ we have

since $\varepsilon + \mu \leq 1 + \lambda$; for $x \in X - U_n$, we have $|h_p(x)| \leq \lambda$ for $1 \leq p \leq n$, whence also

which completes the proof of $(16_n)$.

This being so, the series $\sum_{n=1}^{\infty} \lambda^n h_n$ is normally convergent in $X$ since $\lambda < 1$ and $|h_n(x)| \leq \mu$ for all $n$ and all $x \in X$; let $u$ be its sum, which belongs to $\mathscr{H}$ since $\mathscr{H}$ is closed. By the relation $(14_n)$, we have $u(a) = \sum_{n=1}^{\infty} \lambda^n$; on the other hand if $x \neq a$, there exists an integer $n$ such that $x \notin U_{n+1}$; therefore $|h_{n+k}(x)| \leq \lambda$ for all $k \geq 1$ by the relation $(15_n)$; it follows, using $(16_n)$, that