We first generalize the classical implicit function theorem of Hildebrandt and Graves to the case where we have a Keller -map f defined on an open subset of E×F and with values in F, for E an arbitrary Hausdorff locally convex space and F a Banach space. As an application, we prove that under a certain transversality condition the preimage of a submanifold is a submanifold for a map from a Fréchet manifold to a Banach manifold.
@article{bwmeta1.element.bwnjournal-article-smv134i3p235bwm,
author = {Seppo Hiltunen},
title = {Implicit functions from locally convex spaces to Banach spaces},
journal = {Studia Mathematica},
volume = {133},
year = {1999},
pages = {235-250},
zbl = {0934.58008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv134i3p235bwm}
}
Hiltunen, Seppo. Implicit functions from locally convex spaces to Banach spaces. Studia Mathematica, Tome 133 (1999) pp. 235-250. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv134i3p235bwm/
[00000] [1] A. Frölicher and W. Bucher, z Calculus in Vector Spaces without Norm, Lecture Notes in Math. 30, Springer, Berlin, 1966. | Zbl 0156.38303
[00001] [2] A. Frölicher and A. Kriegl, z Linear Spaces and Differentiation Theory, Pure Appl. Math., Wiley, Chichester, 1988.
[00002] [3] M. W. Hirsch, Differential Topology, Springer, New York, 1976.
[00003] [4] H. H. Keller, z Differential Calculus in Locally Convex Spaces, Lecture Notes in Math. 417, Springer, Berlin, 1974.
[00004] [5] A. Kriegl, Eine kartesisch abgeschlossene Kategorie glatter Abbildungen zwischen beliebigen lokalkonvexen Vektorräumen, Monatsh. Math. 95 (1983), 287-309.
[00005] [6] A. Kriegl and P. W. Michor, z Aspects of the theory of infinite dimensional manifolds, Differential Geom. Appl. 1 (1991), 159-176. | Zbl 0782.58011
[00006] [7] S. Lang, Differential Manifolds, Springer, New York, 1988.
[00007] [8] U. Seip, Kompakt erzeugte Vektorräume und Analysis, Lecture Notes in Math. 273, Springer, Berlin, 1972. | Zbl 0242.46003