Abstract
We prove a generalized implicit function theorem for Banach spaces, without the usual assumption that the subspaces involved being complemented. Then we apply it to the problem of parametrization of fibers of differentiable maps, the Lie subgroup problem for Banach–Lie groups, as well as Weil’s local rigidity for homomorphisms from finitely generated groups to Banach–Lie groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
An J., Wang Z.: Nonabelian cohomology with coefficients in Lie groups. Trans. Am. Math. Soc. 360(6), 3019–3040 (2008)
Benveniste E.J.: Rigidity of isometric lattice actions on compact Riemannian manifolds. Geom. Funct. Anal. 10(3), 516–542 (2000)
Bourbaki, N.: Topological Vector Spaces, Chaps. 1–5. Springer, Berlin (1987)
Bourbaki, N.: Lie Groups and Lie Algebras, Chaps. 1–3. Springer, Berlin (1989)
Browder, F.E.: Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces. In: Proc. Symp. Pure Math. XVIII:2. Am. Math. Soc. (1976)
Borel A., Wallach N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd edn. American Mathematical Society, Providence (2000)
Fisher, D.: First cohomology and local rigidity of group actions. preprint, math.DG/0505520
Glöckner H., Neeb K.-H.: Banach–Lie quotients, enlargibility, and universal complexifications. J. Reine Angew. Math. 560, 1–28 (2003)
Gruenberg K.W.: Resolutions by relations. J. Lond. Math. Soc. 35, 481–494 (1960)
Hamilton R.S.: Deformation of complex structures on manifolds with boundary I: The stable case. J. Differ. Geometry 12(1), 1–45 (1977)
Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. (N.S.) 7(1), 65–222 (1982)
Hofmann, K.H.: Analytic groups without analysis. Symposia Mathematica 16 (Convegno sui Gruppi Topologici e Gruppi di Lie, INDAM, Rome, 1974), pp. 357–374. Academic Press, London (1975)
Neeb, K.-H. (2004) Infinite-dimensional Lie groups and their representations. In: Anker, J.P., Ørsted B. Lie Theory: Lie Algebras and Representations Progress in Math, vol. 228, pp. 213–328. Birkhäuser, Basel (2004)
Neeb K.H.: Towards a Lie theory of locally convex groups. Jap. J. Math. 3rd Ser 1(2), 291–468 (2006)
Omori, H.: Infinite-Dimensional Lie Transformations Groups. Lecture Notes in Math., vol. 427. Springer, Heidelberg (1974)
Rudin, W.: Functional Analysis. McGraw Hill, New York (1973)
Weil A.: Remarks on the cohomology of groups. Ann. Math. 80(2), 149–157 (1964)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
An, J., Neeb, KH. An implicit function theorem for Banach spaces and some applications. Math. Z. 262, 627–643 (2009). https://doi.org/10.1007/s00209-008-0394-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-008-0394-6