Skip to main content Accessibility help
×
Home

AN ALTERNATIVE APPROACH TO FRÉCHET DERIVATIVES

  • SHANE ARORA (a1), HAZEL BROWNE (a2) and DANIEL DANERS (a3)

Abstract

We discuss an alternative approach to Fréchet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carathéodory. The approach allows many questions of differentiability to be reduced to questions of continuity. We demonstrate how that simplifies the theory of differentiation, including the rules of differentiation and the Schwarz lemma on the symmetry of second-order derivatives. We also provide a short proof of the differentiable dependence of fixed points in the Banach fixed point theorem.

Copyright

Corresponding author

References

Hide All
[1]Acosta, E. G. and Delgado, C. G., ‘Fréchet vs. Carathéodory’, Amer. Math. Monthly (4) 101 (1994), 332338.
[2]Adams, J. F., ‘Vector fields on spheres’, Ann. of Math. (2) 75 (1962), 603632.
[3]Amann, H. and Escher, J., Analysis. II (Birkhäuser, Basel, 2008).
[4]Bartle, R. G. and Sherbert, D. R., Introduction to Real Analysis, 4th edn (John Wiley & Sons, Hoboken, NJ, 2011).
[5]Botsko, M. W. and Gosser, R. A., ‘On the differentiability of functions of several variables’, Amer. Math. Monthly (9) 92 (1985), 663665.
[6]Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext (Springer, New York, 2011).
[7]Brooks, R. M. and Schmitt, K., The Contraction Mapping Principle and Some Applications, Electronic Journal of Differential Equations Monograph, 9 (Texas State University – San Marcos, Department of Mathematics, San Marcos, TX, 2009).
[8]Cabrales, R. C. and Rojas-Medar, M. A., ‘Sobre la diferenciabilidad de funciones en espacios de Banach’, Rev. Integr. Temas Mat. (2) 24 (2006), 87100.
[9]Carathéodory, C., Funktionentheorie. Band I (Birkhäuser, Basel, 1950).
[10]Eckmann, B., ‘Stetige Lösungen linearer Gleichungssysteme’, Comment. Math. Helv. 15 (1943), 318339.
[11]Fréchet, M., ‘Sur la notion de différentielle totale’, Nouv. Ann. (4) 12 (1912), 385403.
[12]Hairer, E. and Wanner, G., Analysis by its History (Springer, New York, 2008).
[13]Hale, J. K., Ordinary Differential Equations, Pure and Applied Mathematics, XXI (Wiley-Interscience, New York, 1969).
[14]Henry, D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840 (Springer, Berlin, 1981).
[15]Kuhn, S., ‘The derivative à la Carathéodory’, Amer. Math. Monthly (1) 98 (1991), 4044.
[16]Martínez de la Rosa, F., Cálculo Diferencial: Consideraciones Teóricas y Metodológicas (Universidad de Cádiz, Servicio de Publicaciones, Cádiz, 1998).
[17]Pinzón, S. and Paredes, M., ‘La derivada de Carathéodory en ℝ2’, Rev. Integr. Temas Mat. (2) 17 (2003), 6598.
[18]Rudin, W., Principles of Mathematical Analysis, 3rd edn, International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1976).
[19]Stolz, O., Grundzüge der Differential- und Integralrechnung. Erster Teil: Reelle Veränderliche und Functionen (B. G. Teubner, Leipzig, 1893).
[20]Taylor, A. E. and Lay, D. C., Introduction to Functional Analysis, 2nd edn (John Wiley & Sons, New York, 1980).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

AN ALTERNATIVE APPROACH TO FRÉCHET DERIVATIVES

  • SHANE ARORA (a1), HAZEL BROWNE (a2) and DANIEL DANERS (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.