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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Smooth extensions in infinite dimensional Banach spaces

Author: Peter Renz
Journal: Trans. Amer. Math. Soc. 168 (1972), 121-132
MSC: Primary 58B05
MathSciNet review: 0298712
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Abstract: If $ B$ is $ {l_p}(\omega )$ or $ {c_0}(\omega )$ we show $ B$ has the following extension property. Any homeomorphism from a compact subset $ M$ of $ B$ into $ B$ may be extended to a homeomorphism of $ B$ onto $ B$ which is a $ {C^\infty }$ diffeomorphism on $ B\backslash M$ to its image in $ B$. This is done by writing $ B$ as a direct sum of closed subspaces $ {B_1}$ and $ {B_2}$ both isomorphically isometric to $ B$ so that the natural projection of $ K$ into $ {B_1}$ along $ {B_2}$ is one-to-one (see H. H. Corson, contribution in Symposium on infinite dimensional topology, Ann. of Math. Studies (to appear)). With $ K,B,{B_1}$ and $ {B_2}$ as above a homeomorphism of $ B$ onto itself is constructed which leaves the $ {B_1}$-coordinates of points in $ B$ unchanged, carries $ K$ into $ {B_1}$ and is a $ {C^\infty }$ diffeomorphic map on $ B\backslash K$. From these results the extension theorem may be proved by standard methods.

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Keywords: Extension of homeomorphisms, infinite dimensional Banach spaces, $ {C^\infty }$ extensions, compact subsets of infinite dimensional spaces
Article copyright: © Copyright 1972 American Mathematical Society