Lemma 6

Let $K\subset \mathbf{C}^{\Lambda}$ be a polynomially convex compact subset and U a neighbourhood of K. There exists a finite subset $\Lambda_0$ of Λ such that, for every subset $\Lambda '$ of Λ containing $\Lambda_0$, the set pr$_{\Lambda'}(U)$ contains the polynomially convex hull of pr$_{\Lambda'}(K)$.

Since K is compact, there exists a family of compact disks $D_{\lambda}$ in $\mathbf{C}$ with center 0 and radii $R_{\lambda}$ such that K is contained in the product $D =\prod_{\lambda}D_{\lambda}$. For every $P\in \mathbf{C}[(X_{\lambda})]$, let $K_P$ be the set of

$x\in \mathbf{C}^{\Lambda}$ such that

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

One has

which is contained in U. Thus, the space D is the union of the open set $D\cap U$ and the family of open sets D - $K_P$ for $P\in \mathbf{C}[(X_{\lambda})]$. Since D is compact, there exists an integer $q\geqslant 1$ and polynomials $P_1, . . . ,P_q\in \mathbf{C}[(X_{\lambda})]$ such that:

There exists a finite subset $\Lambda_0\subset$ Λ such that $P_i\in \mathbf{C}[(X_{\lambda})_{\lambda\in\Lambda_0}]$ for $1\leqslant i\leqslant q$. Let us prove that $\Lambda_0$ verifies the assertion of the lemma.

Let $\Lambda '$ be a subset of Λ containing $\Lambda_0$. Let E be the subset of $\mathbf{C}^{\Lambda'}$ formed by the elements $c= (c_{\lambda})_{\lambda\in\Lambda'}$ defined by the inequalities $|c_{\lambda}|\leqslant R_{\lambda}$ for $\lambda \in \Lambda '$, and

$|P_i(c)|\leqslant$ sup$_{c\in K}|P_i(c)|$

for $i= 1, . . . ,q$. The subset E is polynomially convex (lemma 3) and formula (3) proves that pr$_{\Lambda'}(K)\subset E$.

On the other hand, let $c= (c_{\lambda})_{\lambda\in\Lambda'}\in E$; let $d= (d_{\lambda})_{\lambda\in\Lambda}$ be the element of $\mathbf{C}^{\Lambda}$ defined by $d_{\lambda}=c_{\lambda}$ for $\lambda \in \Lambda '$ and $d_{\lambda}= 0$ for $\lambda \in \Lambda$- $\Lambda '$. Then (4) implies that $d\in U$, and hence that $c\in$ pr$_{\Lambda'}(U)$. Thus, $E\subset$ pr$_{\Lambda'}(U)$, which completes the proof.