On spaces of arc-smooth maps
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909000314912768 |
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| author | Rainer, Armin |
| author_facet | Rainer, Armin |
| contents | It is well-known that a function on an open set in $\mathbb R^d$ is smooth if and only if it is arc-smooth, i.e., its composites with all smooth curves are smooth. In recent work, we extended this and related results (for instance, a real analytic version) to suitable closed sets, notably, sets with Hölder boundary and fat subanalytic sets satisfying a necessary topological condition. In this paper, we prove that the resulting set-theoretic identities of function spaces are bornological isomorphisms with respect to their natural locally convex topologies. Extending the results to maps with values in convenient vector spaces, we obtain corresponding exponential laws. Additionally, we show analogous results for special ultradifferentiable Braun-Meise-Taylor classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07023 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On spaces of arc-smooth maps Rainer, Armin Classical Analysis and ODEs Differential Geometry Functional Analysis It is well-known that a function on an open set in $\mathbb R^d$ is smooth if and only if it is arc-smooth, i.e., its composites with all smooth curves are smooth. In recent work, we extended this and related results (for instance, a real analytic version) to suitable closed sets, notably, sets with Hölder boundary and fat subanalytic sets satisfying a necessary topological condition. In this paper, we prove that the resulting set-theoretic identities of function spaces are bornological isomorphisms with respect to their natural locally convex topologies. Extending the results to maps with values in convenient vector spaces, we obtain corresponding exponential laws. Additionally, we show analogous results for special ultradifferentiable Braun-Meise-Taylor classes. |
| title | On spaces of arc-smooth maps |
| topic | Classical Analysis and ODEs Differential Geometry Functional Analysis |
| url | https://arxiv.org/abs/2503.07023 |