Geodesic
A generalization of Steenrod’s approximation theorem
Archivum mathematicum, Tome 45 (2009) no. 2, pp. 95-104

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In this paper we aim for a generalization of the Steenrod Approximation Theorem from [16, Section 6.7], concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalization is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a continuous section in a smooth locally trivial bundles can always be smoothed out in a very controlled way (in terms of the graph topology on spaces of continuous functions), preserving the section on regions where it is already smooth.
Classification : 57R10, 57R12, 58B05
Keywords: infinite-dimensional manifold; infinite-dimensional smooth bundle; smoothing of continuous sections; density of smooth in continuous sections; topology on spaces of continuous functions 
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     author = {Wockel, Christoph},
     title = {A generalization of {Steenrod{\textquoteright}s} approximation theorem},
     journal = {Archivum mathematicum},
     pages = {95--104},
     publisher = {mathdoc},
     volume = {45},
     number = {2},
     year = {2009},
     mrnumber = {2591666},
     zbl = {1212.58005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2009__45_2_a2/}
}
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Wockel, Christoph. A generalization of Steenrod’s approximation theorem. Archivum mathematicum, Tome 45 (2009) no. 2, pp. 95-104. http://geodesic.mathdoc.fr/item/ARM_2009__45_2_a2/