Definition 4

Let $\Lambda$ be a set and V a subset of $\mathbf{C}^{\Lambda}$. The polynomially convex envelope of V is the smallest polynomially convex subset of $\mathbf{C}^{\Lambda}$ containing V.

The polynomially convex envelope of V is the set of $c$ belonging to $\mathbf{C}^{\Lambda}$ such that $|P(c)|\leqslant$ sup$_W|P|$ for all $P\in \mathbf{C}[(X_{\lambda})]$. In fact, this set is polynomially convex by Lemma 3, and is contained in every polynomially convex set containing V by definition.