This is a continuation of an article by

*J. Bonet, L. Frerick* and

*E. Jordá* [Stud. Math. 183, No. 3, 225–248 (2007;

Zbl 1141.46017)]. One of the main theorems reads as follows: Let

${\Omega}$ be an open and connected subset of

${\mathbb{R}}^{N}$ and

$\mathscr{H}$ a sheaf of smooth functions on

${\Omega}$ which is closed in the sheaf

$\mathcal{C}$ of continuous functions on

${\Omega}$; i.e.,

$\mathscr{H}\left(\omega \right)$ is closed in

$\mathcal{C}\left(\omega \right)$ for each open

$\omega \subset {\Omega}$. Let

$M$ be a set of uniqueness for

$\mathscr{H}\left({\Omega}\right)$ (which means that a function in

$\mathscr{H}\left({\Omega}\right)$ which vanishes on

$M$ must vanish on

${\Omega}$) and let

$E$ be a locally convex space. Let

$f:M\to E$ be a function such that

$u\circ f$ has an extension

${f}_{u}\in \mathscr{H}\left({\Omega}\right)$ for each

$u\in W\subset {E}^{\text{'}}$. If (i)

$W$ determines boundedness, or if (ii)

$E$ is Fréchet,

$W={\bigcup}_{n}{B}_{n}$, where

${\left({B}_{n}\right)}_{n}$ fixes the topology of

$E$ (i.e., the polars

${\left({B}_{n}^{\circ}\right)}_{n}$ in

$E$ form a fundamental system of 0-neighbourhoods in

$E$), and

${\left({f}_{u}\right)}_{u\in {B}_{n}}$ is bounded in

$\mathscr{H}\left({\Omega}\right)$ for each

$n$, then

$f$ has an extension

$F\in \mathscr{H}({\Omega},E)$. The case (i) has already been proved in the paper by Bonet and the present authors which is quoted above.