Lemma 3

Let $\Lambda$ be a set. A subset V of $\mathbf{C}^{\Lambda}$ is polynomially convex if and only if there exist a family $(P_i)_{i\in I}$ of elements of $\mathbf{C}[(X_{\lambda})_{\lambda\in\Lambda}]$ and a family $(M_i)_{i\in I}$ of elements of $[0,+\infty ]$ such that V is the set of $c\in \mathbf{C}^{\Lambda}$ satisfying

for all $i\in I$.

If the subset V of $\mathbf{C}^{\Lambda}$ is polynomially convex, it satisfies the above condition for the family consisting of the elements P of $\mathbf{C}[(X_{\lambda})_{\lambda\in\Lambda}]$ by setting $M_P=$ sup$_{c\in V}|P(c)|$.

Conversely, let $(P_i)_{i\in I}$ be a family of elements of $\mathbf{C}[(X_{\lambda})_{\lambda\in\Lambda}]$ and $(M_i)_{i\in I}$ a family of elements of $[0,+\infty ]$. Let V be the set of $c$ in $\mathbf{C}^{\Lambda}$ such that $|P_i(c)|\leqslant M_i$ for all $i\in I$. Then sup$_{c\in V}|P_i(c)|\leqslant M_i$ for $i\in I$. Suppose that $x\in \mathbf{C}^{\Lambda}$ satisfies

$|P(x)|\leqslant$ sup$_{c\in V}|P(c)|$

for all $P\in \mathbf{C}[(X_{\lambda})]$. For $i\in I$, we have in particular $|P_i(x)|\leqslant M_i$, hence $x\in V$. Conversely, for every element $x\in V$ and every polynomial $P\in \mathbf{C}[(X_{\lambda})]$, we have $|P(x)|\leqslant$ sup$_{c\in V}|P(c)|$. Consequently, the set V is polynomially convex.