Bayoumi quasi-differential is not different from Fréchet-differential

Fernando Albiac; José Ansorena

Open Mathematics (2012)

  • Volume: 10, Issue: 3, page 1071-1075
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Fernando Albiac, and José Ansorena. "Bayoumi quasi-differential is not different from Fréchet-differential." Open Mathematics 10.3 (2012): 1071-1075. <http://eudml.org/doc/269754>.

@article{FernandoAlbiac2012,
abstract = {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.},
author = {Fernando Albiac, José Ansorena},
journal = {Open Mathematics},
keywords = {Quasi-Banach space; F-space; Quasi-differentiability (also Bayoumi-differentiability or pq-differentiability); Fréchet differentiability; quasi-Banach space; quasi-differentiability},
language = {eng},
number = {3},
pages = {1071-1075},
title = {Bayoumi quasi-differential is not different from Fréchet-differential},
url = {http://eudml.org/doc/269754},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Fernando Albiac
AU - José Ansorena
TI - Bayoumi quasi-differential is not different from Fréchet-differential
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 1071
EP - 1075
AB - 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.
LA - eng
KW - Quasi-Banach space; F-space; Quasi-differentiability (also Bayoumi-differentiability or pq-differentiability); Fréchet differentiability; quasi-Banach space; quasi-differentiability
UR - http://eudml.org/doc/269754
ER -

References

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  1. [1] Albiac F., The role of local convexity in Lipschitz maps, J. Convex. Anal., 2011, 18(4), 983–997 Zbl1236.46002
  2. [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. [3] Bayoumi A., Mean value theorem for complex locally bounded spaces, Comm. Appl. Nonlinear Anal., 1997, 4(4), 91–103 Zbl0910.46032
  4. [4] Bayoumi A., Foundations of Complex Analysis in Non Locally Convex Spaces, North-Holland Math. Stud., 193, Elsevier, Amsterdam, 2003 
  5. [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 Zbl1107.58002
  6. [6] Benyamini Y., Lindenstrauss J., Geometric nonlinear functional analysis. I, Amer. Math. Soc. Colloq. Publ., 48, American Mathematical Society, Providence, 2000 Zbl0946.46002
  7. [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 Zbl0214.28402
  8. [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 Zbl0454.46001
  9. [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 Zbl1059.46004
  10. [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. [11] Rolewicz S., On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Cl. III, 1957, 5, 471–473 Zbl0079.12602
  12. [12] Rolewicz S., Remarks on functions with derivative zero, Wiadom. Mat., 1959, 3, 127–128 (in Polish) 
  13. [13] Rolewicz S., Metric linear spaces, Math. Appl. (East European Ser.), 20, Reidel, Dordrecht, PWN, Warsaw, 1985 

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