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A generalization of Steenrod’s approximation theorem. (English) Zbl 1212.58005
Summary: In this paper we aim for a generalization of the Steenrod approximation theorem from Section 6.7 in [J. Steenrod, The topology of fibre bundles. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press (1999; Zbl 0942.55002)] 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.

58B05 Homotopy and topological questions for infinite-dimensional manifolds
57R10 Smoothing in differential topology
57R12 Smooth approximations in differential topology
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