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2012 | 10 | 3 | 1071-1075
Tytuł artykułu

Bayoumi quasi-differential is not different from Fréchet-differential

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Unlike for Banach spaces, the differentiability of functions between infinite-dimensional nonlocally convex spaces has not yet been properly studied or understood. In a paper published in this Journal in 2006, Bayoumi claimed to have discovered a new notion of derivative that was more suitable for all F-spaces including the locally convex ones with a wider potential in analysis and applied mathematics than the Fréchet derivative. The aim of this short note is to dispel this misconception, since it could hinder making headway in this already hard enough subject. To that end we show that Bayoumi quasi-differentiability, when properly defined, is the same as Fréchet differentiability, and that some of his alleged applications are wrong.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
3
Strony
1071-1075
Opis fizyczny
Daty
wydano
2012-06-01
online
2012-03-24
Twórcy
Bibliografia
  • [1] Albiac F., The role of local convexity in Lipschitz maps, J. Convex. Anal., 2011, 18(4), 983–997
  • [2] Aoki T., Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo, 1942, 18, 588–594 http://dx.doi.org/10.3792/pia/1195573733
  • [3] Bayoumi A., Mean value theorem for complex locally bounded spaces, Comm. Appl. Nonlinear Anal., 1997, 4(4), 91–103
  • [4] Bayoumi A., Foundations of Complex Analysis in Non Locally Convex Spaces, North-Holland Math. Stud., 193, Elsevier, Amsterdam, 2003
  • [5] Bayoumi A., Bayoumi quasi-differential is different from Fréchet-differential, Cent. Eur. J. Math., 2006, 4(4), 585–593 http://dx.doi.org/10.2478/s11533-006-0028-3
  • [6] Benyamini Y., Lindenstrauss J., Geometric nonlinear functional analysis. I, Amer. Math. Soc. Colloq. Publ., 48, American Mathematical Society, Providence, 2000
  • [7] Enflo P., Uniform structures and square roots in topological groups. I, Israel J. Math., 1970, 8, 230–252 http://dx.doi.org/10.1007/BF02771560
  • [8] Kalton N.J., Curves with zero derivatives in F-spaces, Glasgow Math. J., 1981, 22(1), 19–29 http://dx.doi.org/10.1017/S0017089500004432
  • [9] Kalton N., Quasi-Banach spaces, In: Handbook of the Geometry of Banach Spaces, 2, North-Holland, Amsterdam, 2003, 1099–1130 http://dx.doi.org/10.1016/S1874-5849(03)80032-3
  • [10] Kalton N.J., Peck N.T., Roberts J.W., An F-space Sampler, London Math. Soc. Lecture Note Ser., 89, Cambridge University Press, 1984
  • [11] Rolewicz S., On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Cl. III, 1957, 5, 471–473
  • [12] Rolewicz S., Remarks on functions with derivative zero, Wiadom. Mat., 1959, 3, 127–128 (in Polish)
  • [13] Rolewicz S., Metric linear spaces, Math. Appl. (East European Ser.), 20, Reidel, Dordrecht, PWN, Warsaw, 1985
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0031-9
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