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Implicit functions from topological vector spaces to Banach spaces

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Abstract

We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.

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References

  1. N. A 'Campo,Théorème de préparation différentiable ultra-métrique, Séminaire Delange-Pisot-Poitou 1967/68, Théorie des Nombres, Fasc. 2, Exp. 17, (1969), 7 pp. Secrétariat mathématique, Paris.

  2. W. Bertram, H. Glöckner and K.-H. Neeb,Differential calculus over general base fields and rings, Exposiones Mathematicae22 (2004), 213–282.

    Article  MATH  Google Scholar 

  3. H. Biller,Analyticity and naturality of the multi-variable functional calculus, TU Darmstadt Preprint2332, April 2004.

  4. J. Bochnak and J. Siciak,Analytic functions in topological vector spaces, Studia Mathematica39 (1971), 77–112.

    MATH  MathSciNet  Google Scholar 

  5. N. Boja,A survey on non-archimedian immersions, Filomat12 (1998), 31–51.

    MATH  MathSciNet  Google Scholar 

  6. N. Bourbaki,Variétés différentielles et analytiques.Fascicule de résultats, Hermann, Paris, 1967.

    MATH  Google Scholar 

  7. N. Bourbaki,Topological Vector Spaces, (Chapters 1–5), Springer, Berlin, 1987.

    MATH  Google Scholar 

  8. H. Cartan,Calcul différentiel, Hermann, Paris, 1967.

    MATH  Google Scholar 

  9. J. Dieudonné,Foundations of Modern Analysis, Academic Press, New York and London, 1960.

    MATH  Google Scholar 

  10. A. Frölicher and A. Kriegl,Linear Spaces and Differentiation Theory, Wiley, New York, 1988.

    MATH  Google Scholar 

  11. H. Glöckner,Scale functions on p-adic Lie groups, Manuscripta Mathematica97 (1998), 205–215.

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Glöckner,Infinite-dimensional Lie groups without completeness restrictions, inGeometry and Analysis on Finite- and Infinite-dimensional Lie Groups (A. Strasburger et al., eds.), Banach Center Publications55, Warsaw, 2002, pp. 43–59.

  13. H. Glöckner,Lie group, structures on quotient groups and universal complexifications for infinite-dimensional Lie groups, Journal of Functional Analysis194 (2002), 347–409.

    Article  MATH  MathSciNet  Google Scholar 

  14. H. Glöckner,Smooth Lie groups over local fields of positive characteristic need not be analytic, Journal of Algebra285 (2005), 356–371.

    Article  MATH  MathSciNet  Google Scholar 

  15. H. Glöckner,Every smooth p-adic Lie group admits a compatible analytic structure, Forum Mathematicum, to appear (also arXiv:math.GR/0312113).

  16. H. Glöckner,Lie groups over non-discrete topological fields, preprint, arXiv:math.GR/0408008.

  17. H. Glöckner,Bundles of locally convex spaces, group actions, and hypocontinuons bilinear mappings, submitted (will be made available as a TU Darmstadt preprint).

  18. H. Glöckner,Stable manifolds for dynamical systems over ultrametric fields, in preparation.

  19. H. Glöckner,Pseudo-stable manifolds for dynamical systems over ultrametric fields, in preparation.

  20. H. Glöckner,Scale functions on Lie Groups over local fields of positive characteristic, in preparation.

  21. H. Glöckner and K.-H. Neeb,Infinite-Dimensional Lie Groups, Vol. I, book in preparation.

  22. R. Hamilton,The inverse function theorem of Nash and Moser, Bulletin of the American Mathematical Society7 (1982), 65–222.

    MATH  MathSciNet  Google Scholar 

  23. T. H. Hildebrandt and L. M. Graves,Implicit functions and their differentials in general analysis, Transactions of the American Mathematical Society29 (1927), 127–153.

    Article  MathSciNet  Google Scholar 

  24. S. Hiltunen,Implicit functions from locally convex spaces to Banach spaces, Studia Mathematica,134 (1999), 235–250.

    MATH  MathSciNet  Google Scholar 

  25. S. Hiltunen,A Frobenius theorem for locally convex global analysis, Monatshefte für Mathematik129 (2000), 109–117.

    Article  MATH  MathSciNet  Google Scholar 

  26. S. Hiltunen,Differentiation, implicit functions, and applications of generalized well-posedness, preprint, arXiv:math.FA/0504268.

  27. M. C. Irwin,On the stable manifold theorem, The Bulletin of the London Mathematical Society2 (1970), 196–198.

    Article  MATH  MathSciNet  Google Scholar 

  28. M. C. Irwin,A new proof of the speudostable manifold theorem, Journal of the London Mathematical Society21 (1980), 557–566.

    Article  MATH  MathSciNet  Google Scholar 

  29. H. H. Keller,Differential Calculus in Locally Convex Spaces, Springer, Berlin, 1974.

    MATH  Google Scholar 

  30. S. G. Krantz and H. R. Parks,The Implicit Function Theorem, Birkhäuser, Basel, 2002.

    MATH  Google Scholar 

  31. A. Kriegl and P. W. Michor,The Convenient Setting of Global Analysis, American Mathematical Society, Providence, RI, 1997.

    MATH  Google Scholar 

  32. S. Lang,Fundamentals of Differential Geometry, Springer, Berlin, 1999.

    MATH  Google Scholar 

  33. E. B. Leach,A Note on inverse function theorems, Proceedings of the American Mathematical Society12 (1961), 694–697.

    Article  MATH  MathSciNet  Google Scholar 

  34. E. B. Leach,On a related function theorem, Proceedings of the American Mathematical Society14 (1963), 687–689.

    Article  MATH  MathSciNet  Google Scholar 

  35. S. V. Ludkovsky,Measures on groups of diffeomorphisms of non-archimedian Banach manifolds, Russian Mathematical Surveys51 (1996), 338–340.

    Article  MathSciNet  Google Scholar 

  36. S. V. Ludkovsky,Quasi-invariant measures on non-Archimedian groups and semigroups of loops and paths, their representations I, Annales Mathématiques Blaise Pascal7 (2000), 19–53.

    MATH  MathSciNet  Google Scholar 

  37. T. W. Ma,Inverse mapping theorem on coordinate spaces, The Bulletin of the London Mathematical Society33 (2001), 473–482.

    MATH  MathSciNet  Google Scholar 

  38. P. W. Michor,Manifolds of Differentiable Mappings, Shiva Publishing, Orpington, 1980.

    MATH  Google Scholar 

  39. J. Milnor,Remarks on infinite-dimensional Lie groups, inRelativity, Groups and Topology II (B. DeWitt and R. Stora, eds.), North-Holland, Amsterdam, 1983, pp. 1008–1057.

    Google Scholar 

  40. A. F. Monna,Analyse Non-Archimédienne, Springer, Berlin, 1979.

    Google Scholar 

  41. K.-H. Neeb,Central extensions of infinite-dimensional Lie groups, Annales de l'Institut Fourier (Grenoble)52 (2002), 1365–1442.

    MATH  MathSciNet  Google Scholar 

  42. A. C. M. Rooij,Non-Archimedian Functional Analysis, Marcel Dekker, New York, 1978.

    Google Scholar 

  43. W. H. Schikhof,Ultrametric Calculus, Cambridge University Press, 1984.

  44. J.-P. Serre,Lie Algebras and Lie Groups, Springer, Berlin, 1992.

    MATH  Google Scholar 

  45. B. Slezák,On the inverse function theorem and implicit function theorem in Banach spaces, inFunction Spaces (Poznań, 1986) (J. Musielak, ed.), Teubner, Leipzig, 1988, pp. 186–190.

    Google Scholar 

  46. S. De Smedt,Local invertibility of non-archimedian vector-valued functions, Annales Mathématiques Blaise Pascal5 (1998), 13–23.

    MATH  Google Scholar 

  47. J. Teichmann,A Frobenius theorem on convenient manifolds, Monatshefte für Mathematik134 (2001), 159–167.

    Article  MATH  MathSciNet  Google Scholar 

  48. J. S. P. Wang,The Mautner phenomenon for p-adic Lie groups, Mathematische Zeitschrift185 (1985), 403–412.

    Article  Google Scholar 

  49. A. Weil,Basic Number Theory, Springer, Berlin, 1973.

    MATH  Google Scholar 

  50. W. Więsŀaw,Topological Fields, Marcel Dekker, New York and Basel, 1988.

    Google Scholar 

  51. G. A. Willis,The structure of totally disconnected, locally compact groups, Mathematische Annalen300 (1994), 341–363.

    Article  MATH  MathSciNet  Google Scholar 

  52. G. A. Willis,Further properties of the scale function on a totally disconnected group, Journal of Algebra237 (2001), 142–164.

    Article  MATH  MathSciNet  Google Scholar 

  53. T. Wurzbacher,Fermionic second quantization and the geometry of the restricted Grassmannian, inInfinite-Dimensional Kähler Manifolds (A. T. Huckleberry and T. Wurzbacher, eds.), DMV-Seminar31, Birkhäuser-Verlag, Basel, 2001, pp. 287–375.

    Google Scholar 

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  1. TU Darmstadt, FB Mathematik AG 5, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Helge Glöckner

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  1. Helge Glöckner
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Glöckner, H. Implicit functions from topological vector spaces to Banach spaces. Isr. J. Math. 155, 205–252 (2006). https://doi.org/10.1007/BF02773955

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