A tame splitting theorem for exact sequences of Fréchet spaces
The following theorem is proved: Let \(F\), \(G\), \(E\) be Fréchet spaces which are equipped with fundamental systems of Hilbertian seminorms \(\|\;\|_ 0\leq \|\;\|_ 1\leq\dots\) Assume that there are an integer \(p_ 0\), real numbers \(0< \theta_ k< 1\), \(0< \tau_ k< 1\) with \(\theta_ k\geq \tau_{k+ p}\) for all \(p\geq p_ 0\) and all \(k\), and constants \(c_ k> 0\) such that (1) \(\|\;\|^*_ k\leq c_ k\|\;\|^{* 1-\theta_ k}_{k- 1} \|\;\|^{*\theta_ k}_{k+ 1}\) on \((F, \|\;\|_{k- 1})'\), (2) \(\|\;\|_ k\leq c_ k\|\;\|^{1- \tau_ k}_{k- 1} \|\;\|^{\tau_ k}_{k+ 1}\) on \(E\) for all \(k\). Then every tamely exact sequence \[ 0\longrightarrow F\longrightarrow G\longrightarrow E\longrightarrow 0 \] splits tamely. Here \(\|\;\|^*_ k\) denotes the dual norm \(\|\phi \|^*_ k= \sup\{| \phi(x)|: \| x\|_ k\leq 1\}\), \(\phi\in F'\), and a linear map \(T: E\to F\) is called tame if \(\| Tx\|_ n\leq c_ n\| x\|_{n+ b}\) for a fixed \(b\) and all \(n\). As an application we give a short proof of a result of Mityagin and Heinkin on the existence of continuous linear extension operators for holomorphic functions on complex manifolds in strictly pseudoconvex regions in \(\mathbb{C}^ N\).
fundamental systems of Hilbertian seminorms
existence of continuous linear extension operators
holomorphic functions on complex manifolds in strictly pseudoconvex regions
scientific article; zbMATH DE number 784105
scientific article