# Article

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Keywords:
infinite-dimensional manifold; infinite-dimensional smooth bundle; smoothing of continuous sections; density of smooth in continuous sections; topology on spaces of continuous functions
Summary:
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.
References:
[1] Bourbaki, N.: General topology. Elements of Mathematics (Berlin) (1998), Springer-Verlag, Berlin, translated from the French. Zbl 0894.54001
[2] Dugundji, J.: Topology. Allyn and Bacon Inc., Boston, 1966. MR 0193606 | Zbl 0144.21501
[3] Glöckner, H.: Infinite-dimensional Lie groups without completeness restrictions. Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups (Bȩdlewo, 2000), vol. 55, Banach Center Publ., 43–59, Polish Acad. Sci., Warsaw, 2002. MR 1911979 | Zbl 1020.58009
[4] Glöckner, H., Neeb, K.-H.: Infinite-dimensional Lie groups. Basic Theory and Main Examples, volume I, Springer-Verlag, 2009, in preparation.
[5] Hamilton, R. S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7 (1) (1982), 65–222. DOI 10.1090/S0273-0979-1982-15004-2 | MR 0656198 | Zbl 0499.58003
[6] Hirsch, M. W.: Differential Topology. Springer-Verlag, New York, 1976. MR 0448362 | Zbl 0356.57001
[7] Keller, H. H.: Differential calculus in locally convex spaces. Lecture Notes in Math., vol. 417, Springer-Verlag, Berlin, 1974. MR 0440592 | Zbl 0293.58001
[8] Kriegl, A., Michor, P. W.: Smooth and continuous homotopies into convenient manifolds agree. unpublished preprint, 2002, available from http://www.mat.univie.ac.at/\$\sim \$michor/.
[9] Kriegl, A., Michor, P. W.: The Convenient Setting of Global Analysis. Math. Surveys Monogr., vol. 53, Amer. Math. Soc., 1997. MR 1471480 | Zbl 0889.58001
[10] Lee, J. M.: Introduction to smooth manifolds. Grad. Texts in Math., vol. 218, Springer-Verlag, New York, 2003. DOI 10.1007/978-0-387-21752-9 | MR 1930091
[11] Michor, P. W.: Manifolds of Differentiable Mappings. Shiva Mathematics Series, vol. 3, Shiva Publishing Ltd., Nantwich, 1980, out of print, online available from http://www.mat.univie.ac.at/\$\sim \$michor/. MR 0583436 | Zbl 0433.58001
[12] Milnor, J.: Remarks on infinite-dimensional Lie groups. Relativity, Groups and Topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984, pp. 1007–1057. MR 0830252 | Zbl 0594.22009
[13] Müller, C., Wockel, C.: Equivalences of smooth and continuous principal bundles with infinite-dimensional structure group. Adv. Geom., to appear, 2009, arXiv:math/0604142.
[14] Naimpally, S. A.: Graph topology for function spaces. Trans. Amer. Math. Soc. 123 (1966), 267–272. DOI 10.1090/S0002-9947-1966-0192466-4 | MR 0192466 | Zbl 0151.29703
[15] Neeb, K.-H.: Central extensions of infinite-dimensional Lie groups. Ann. Inst. Fourier (Grenoble) 52 (5) (2002), 1365–1442. DOI 10.5802/aif.1921 | MR 1935553 | Zbl 1019.22012
[16] Steenrod, N.: The Topology of Fibre Bundles. Princeton Math. Ser. 14 (1951). MR 0039258 | Zbl 0054.07103
[17] Wockel, C.: Smooth extensions and spaces of smooth and holomorphic mappings. J. Geom. Symmetry Phys. 5 (2006), 118–126. MR 2269885 | Zbl 1108.58006

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