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Extension of vector-valued functions. (English) Zbl 1141.46018
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 ${ℝ}^{N}$ and $ℋ$ a sheaf of smooth functions on ${\Omega }$ which is closed in the sheaf $𝒞$ of continuous functions on ${\Omega }$; i.e., $ℋ\left(\omega \right)$ is closed in $𝒞\left(\omega \right)$ for each open $\omega \subset {\Omega }$. Let $M$ be a set of uniqueness for $ℋ\left({\Omega }\right)$ (which means that a function in $ℋ\left({\Omega }\right)$ 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 ${f}_{u}\in ℋ\left({\Omega }\right)$ 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 $ℋ\left({\Omega }\right)$ for each $n$, then $f$ has an extension $F\in ℋ\left({\Omega },E\right)$. The case (i) has already been proved in the paper by Bonet and the present authors which is quoted above.
##### MSC:
 46E40 Spaces of vector- and operator-valued functions 46A04 Locally convex Fréchet spaces, etc. 46M05 Tensor products of topological linear spaces