Let f be a smooth function defined on a finite union U of open convex sets in a locally convex Lindelöf space E. If, for every x ∈ U, the restriction of f to a suitable neighbourhood of x admits a smooth extension to the whole of E, then the restriction of f to a union of convex sets that is strictly smaller than U also admits a smooth extension to the whole of E.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm146-3-1,
author = {C. J. Atkin},
title = {Extension of smooth functions in infinite dimensions, I: unions of convex sets},
journal = {Studia Mathematica},
volume = {147},
year = {2001},
pages = {201-226},
zbl = {0979.46054},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm146-3-1}
}
C. J. Atkin. Extension of smooth functions in infinite dimensions, I: unions of convex sets. Studia Mathematica, Tome 147 (2001) pp. 201-226. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm146-3-1/