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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
DOI: https://doi.org/10.1090/S0002-9947-1972-0298712-3
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, <!– MATH ${C^\infty }$ –> <IMG WIDTH="38" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${C^\infty }$"> extensions, compact subsets of infinite dimensional spaces
Article copyright: © Copyright 1972 American Mathematical Society