ULTRAMETRIC AND NON-LOCALLY CONVEX ANALOGUES OF THE GENERAL CURVE LEMMA OF CONVENIENT DIFFERENTIAL CALCULUS

GLÖCKNER, HELGE

Glasgow mathematical journal, Tome 50 (2008) no. 2, pp. 271-288

Résumé

The General Curve Lemma is a tool of Infinite-Dimensional Analysis that enables refined studies of differentiability properties of maps between real locally convex spaces to be made. In this article, we generalize the General Curve Lemma in two ways. First, we remove the condition of local convexity in the real case. Second, we adapt the lemma to the case of curves in topological vector spaces over ultrametric fields.

Détail

DOI : 10.1017/S0017089508004199

Mots-clés : 26E15, 26E20, 26E30, 45T20, 46A16, 46S10

Bibliographie

[1] 1.Adasch, N., Ernst, B. and Keim, D., Topological vector spaces. The theory without convexity conditions, (Springer-Verlag, 1978). Google Scholar

[2] 2.Bertram, W., Differential geometry, Lie groups and symmetric spaces over general base fields and rings, Memoirs of the AMS (2008), no. 900, 186+v pp. Google Scholar | DOI

[3] 3.Bertram, W., Glöckner, H. and Neeb, K.-H., Differential calculus over general base fields and rings, Expo. Math. 22 (2004), 213–282. Google Scholar | DOI

[4] 4.Boman, J.Differentiability of a function and of its compositions with functions of one variable, Math Scand. 20 (1967), 249–268. Google Scholar | DOI

[5] 5.Bourbaki, N.Topological vector spaces Chapters 1–5, (Springer-Verlag, 1987). Google Scholar | DOI

[6] 6.De Smedt, S.p-adic continuously differentiable functions of several variables, Collect. Math. 45 (1994), 137–152. Google Scholar

[7] 7.Frölicher, A., and Kriegl, A.Linear spaces and differentiation theory, (John Wiley, 1988). Google Scholar

[8] 8.Glöckner, H.Smooth Lie groups over local fields of positive characteristic need not be analytic, J. Algebra 285 (2005), 356–371. Google Scholar | DOI

[9] 9.Glöckner, H.Every smooth p-adic Lie group admits a compatible analytic structure, Forum Math. 18 (2006), 45–84. Google Scholar | DOI

[10] 10.Glöckner, H., Aspects of p-adic non-linear functional analysis, Khrennikov, A. Yu., Rakić, Z. and Volovich, I. V. (Eds.), p-Adic Mathematical Physics. 2nd International Conference (Belgrade, 2005), AIP Conference Proceedings 826, (Amer. Inst. Physics, New York, 2006) (cf. arXiv:math/0602081), 237–253. Google Scholar

[11] 11.Glöckner, H.Comparison of some notions of Ck-maps in multi-variable non-archimedian analysis, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), 877–904. Google Scholar | DOI

[12] 12.Glöckner, H., Lie groups over non-discrete topological fields, preprint, arXiv:math/0408008. Google Scholar

[13] 13.Glöckner, H., Finite order differentiability properties, fixed points and implicit functions over valued fields, preprint, arXiv:math/0511218. Google Scholar

[14] 14.Glöckner, H., and Neeb, K.-H., Infinite-dimensional Lie groups, Vol. I; book in preparation. Google Scholar

[15] 15.Jarchow, H.Locally convex spaces, (B. G. Teubner, Stuttgart, 1981). Google Scholar | DOI

[16] 16.Kalton, N. J., Peck, N. T. and Roberts, J. W.An F-space sampler, (Cambridge University Press, 1984). Google Scholar | DOI

[17] 17.Kriegl, A., and Michor, P. W., The convenient setting of global analysis, Amer. Math. Soc., (Providence, 1997). Google Scholar

[18] 18.Ludkovsky, S. V.Irreducible unitary representations of non-Archimedean groups of diffeomorphisms, Southeast Asian Bull. Math. 22 (1998), 419–436. Google Scholar

[19] 19.Ludkovsky, S. V.Quasi-invariant measures on non-Archimedean groups and semigroups of loops and paths, their representations I, II Ann. Math. Blaise Pascal 7 (2000), 19–53 and 55–80. Google Scholar | DOI

[20] 20.Ludkovsky, S. V., Smoothness of functions global and along curves over ultra-metric fields, arXiv:math/0608725. Google Scholar

[21] 21.Schikhof, W. H.Ultrametric calculus, (Cambridge University Press, 1984). Google Scholar