{"paper":{"title":"Smooth approximations without critical points of continuous mappings between Banach spaces, and diffeomorphic extractions of sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.FA","authors_text":"Daniel Azagra, Miguel Garc\\'ia-Bravo, Tadeusz Dobrowolski","submitted_at":"2018-11-19T10:08:31Z","abstract_excerpt":"Let $E$, $F$ be separable Hilbert spaces, and assume that $E$ is infinite-dimensional. We show that for every continuous mapping $f:E\\to F$ and every continuous function $\\varepsilon: E\\to (0, \\infty)$ there exists a $C^{\\infty}$ mapping $g:E\\to F$ such that $\\|f(x)-g(x)\\|\\leq\\varepsilon(x)$ and $Dg(x):E\\to F$ is a surjective linear operator for every $x\\in E$. We also provide a version of this result where $E$ can be replaced with a Banach space from a large class (including all the classical spaces with smooth norms, such as $c_0$, $\\ell_p$ or $L^{p}$, $1<p<\\infty$), and $F$ can be taken to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.07587","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}