×

Spaces of Whitney jets on self-similar sets. (English) Zbl 1316.46024

Let \(E\) be a complemented subspace of the Fréchet space \(s\). The author gives a sufficient condition under which \(E\) is isomorphic to \(s\).
More precisely, if a normwise exact sequence \(0 \to E \to E \oplus \ldots \oplus E \to G \to 0\) exists (for \(G=\{0\}\) or \(G=\omega\)), then the Kolmogorov diameters of the local Banach spaces of \(E\) can be estimated and the isomorphism follows by a result in [A. Aytuna et al., Math. Ann. 283, No. 2, 193–202 (1989; Zbl 0643.46001)].
This condition is applied to spaces of Whitney jets on the Cantor set and on the Sierpiński triangle. The results can probably be extended to other self-similar sets.

MSC:

46E10 Topological linear spaces of continuous, differentiable or analytic functions
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A63 Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces

Citations:

Zbl 0643.46001
PDFBibTeX XMLCite
Full Text: DOI Link