Geodesic
On the extension and smoothing of vector-valued functions
Izvestiya. Mathematics , Tome 59 (1995) no. 4, pp. 847-879.

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Smoothing of maps in Banach spaces is considered in this article. We construct an example of an infinitely differentiable vector-valued function on a subspace $L$ in $C[0,1]$ that does not have a uniformly continuous extension to a neighbourhood of $L$. The Kolmogorov widths obtained are correct in the order of growth of three parameters. 
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I. G. Tsar'kov. On the extension and smoothing of vector-valued functions. Izvestiya. Mathematics , Tome 59 (1995) no. 4, pp. 847-879. http://geodesic.mathdoc.fr/item/IM2_1995_59_4_a9/

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