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Summary: arXiv:0906.2989v2[math.FA]28Jan2010
Smooth extensions of functions on separable
Banach spaces
D. Azagra, R. Fry, and L. Keener
Abstract. Let X be a Banach space with a separable dual X
. Let
Y X be a closed subspace, and f : Y R a C1
-smooth function.
Then we show there is a C1
extension of f to X.
1. Introduction
In this note we address the problem of the extension of smooth functions
from subsets of Banach spaces to smooth functions on the whole space.
For our results, smoothness is meant in the Fr´echet sense, and we shall
restrict our attention to real-valued functions. To state the problem more
precisely, given a Banach space X, a closed subset Y, and a Cp-smooth
function f : Y R, when is it possible to find a Cp-smooth map F : X R
such that F |Y = f?
We should note that when Y is a complemented subspace of an arbitrary
Banach space X, the extension problem can be easily solved. Indeed, let
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