Restriction of coherent sections to a relatively compact open set is compact

If ${\mathrm{\Omega }}^{\mathrm{\prime }}=\mathrm{\varnothing }$, the target section space is zero and restriction is the zero compact map. Otherwise, since $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$ is compact in $\mathrm{\Omega }$, choose a compact set $L$ with $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}\subseteq \mathrm{int}L\subseteq L\subseteq \mathrm{\Omega }$. The set $B=\{f\in \mathcal{O}(\mathrm{\Omega }{)}^p:{\sup }_L\mathrm{\parallel }f\mathrm{\parallel }<1\}$ is an open neighborhood of zero in the compact-convergence topology. Cover $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$ by finitely many smaller polydiscs whose slightly larger closed polydiscs lie in $\mathrm{int}L$. By [F2], the family $B$ has uniformly bounded first derivatives on each smaller polydisc, hence is equicontinuous and uniformly bounded on each compact subset of ${\mathrm{\Omega }}^{\mathrm{\prime }}$.

For integers $j\ge 1$, use the compact exhaustion $K_j=\{z\in {\mathrm{\Omega }}^{\mathrm{\prime }}:\mathrm{\mid }z\mathrm{\mid }\le j,\mathrm{dist}(z,{\mathbb{C}}^m\setminus {\mathrm{\Omega }}^{\mathrm{\prime }})\ge 1\mathrm{/}j\}$, with distance to the empty set taken as $+\mathrm{\infty }$. Each $K_j$ is compact by [F6], and their union is ${\mathrm{\Omega }}^{\mathrm{\prime }}$. On each nonempty $K_j$, the family from step 1.1 is equicontinuous and pointwise relatively compact: its values lie in a bounded subset of ${\mathbb{C}}^p$, whose closure is compact by [F6] (for $p=0$ the target is the singleton). Apply [F3] and take nested subsequences diagonally; every sequence in the restrictions of $B$ then has a subsequence converging uniformly on every $K_j$. Its limit is holomorphic by [F4]. By [F1] the compact-convergence topology on $\mathcal{O}({\mathrm{\Omega }}^{\mathrm{\prime }}{)}^p$ is metrizable, so this sequential compactness proves that the restricted set has compact closure. Thus restriction $\mathcal{O}(\mathrm{\Omega }{)}^p\to \mathcal{O}({\mathrm{\Omega }}^{\mathrm{\prime }}{)}^p$ sends the neighborhood $B$ to a set with compact closure.

Let $Q$ and $Q^{\mathrm{\prime }}$ be the quotient maps from the finite free section spaces onto $\mathrm{\Gamma }(\mathrm{\Omega },\mathcal{F})$ and $\mathrm{\Gamma }({\mathrm{\Omega }}^{\mathrm{\prime }},\mathcal{F})$. The image $N=Q(B)$ is a neighborhood of zero because a quotient map is open. Every section in $N$ has a lift in $B$, so the restriction identity ${\mathrm{res}}_{\mathcal{F}}\circ Q=Q^{\mathrm{\prime }}\circ {\mathrm{res}}_{\mathcal{O}}$ puts ${\mathrm{res}}_{\mathcal{F}}(N)$ inside $Q^{\mathrm{\prime }}({\mathrm{res}}_{\mathcal{O}}(B))$. The latter has compact closure, being a continuous image of the compact closure from step 2.1. Thus the coherent-section restriction is compact.