On the domain of implicit functions in a projective limit setting without additional norm estimates

• Autorzy: Magnot Jean-Pierre

Identyfikatory

DOI

10.1515/dema-2020-0008

• Języki publikacji: EN

Abstrakty

EN

We examine how implicit functions on ILB-Fréchet spaces can be obtained without metric or norm estimates which are classically assumed. We obtain implicit functions defined on a domain D which is not necessarily open, but which contains the unit open ball of a Banach space. The corresponding inverse function theorem is obtained, and we finish with an open question on the adequate (generalized) notion of differentiation, needed for the corresponding version of the Fröbenius theorem.

Słowa kluczowe

EN

implicit function; ILB spaces; Fréchet space

PL

funkcja uwikłana; przestrzeń Frécheta

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2020
• Tom: Vol. 53, nr 1
• Strony: 112--120
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Magnot Jean-Pierre

• LAREMA
• University of Angers, Boulevard Lavoisier, F-49045 Angers Cedex 01, France
• Jeanne d'Arc High School, Avenue de Grande Bretagne, F-63000 Clermont-Ferrand, France

Bibliografia

• [1] J. Dieudonné, Foundations of Modern Analysis, enlarged and corrected printing, Pure and Applied Mathematics, Vol. 10-I, Academic Press, New York-London, 1969.
• [2] S. G. Krantz and H. Parks, The Implicit Function Theorem. History, Theory, and Applications, Reprint of the 2003 edition, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2013.
• [3] J.-P. Penot, Sur le théorème de Frobenius, Bull. Soc. Math. France 98 (1970), 47-80.
• [4] H. Glöckner, Implicit functions from topological vector spaces to Banach spaces, Israel J. Math. 155 (2006), 205-252.
• [5] H. Hogbe-Nlend, Théorie des bornologies et applications, Lecture Notes in Mathematics, vol. 213, Springer-Verlag, Berlin New York, 1971.
• [6] A. Kriegl and P. W. Michor, The Convenient Setting for Global Analysis, Mathematical Surveys and Monographs, vol. 53, American Mathematical Society, Providence, RI, 1997.
• [7] H. Omori, Infinite dimensional Lie groups, Translations of Mathematical Monographs, vol. 158, American Mathematical Society, Providence, RI, 1997.
• [8] C. T. J. Dodson, G. Galanis, and E. Vassiliou, Geometry in a Fréchet Context: A Projective Limit Approach, London Mathematical Society Lecture Note Series, vol. 428, Cambridge University Press, Cambridge, 2016.
• [9] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65-222.
• [10] P. Iglesias-Zemmour, Diffeology, Mathematical Surveys and Monographs, vol. 185, American Mathematical Society, Providence, RI, 2013.
• [11] J.-P. Magnot, Ambrose-Singer theorem on diffeological bundles and complete integrability of the KP equation, Int. J. Geom. Methods Mod. Phys. 10 (2013), no. 9, 1350043, DOI: 10.1142/S0219887813500436.
• [12] J.-P. Magnot, On the differential geometry of numerical schemes and weak solutions of functional equations, preprint arXiv:1607.02636.
• [13] J.-P. Magnot and J. Watts, The diffeology of Milnor's classifying space, Topol. Appl. 232 (2017), 189-213.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).