Abstract and Applied Analysis
Volume 2005 (2005), Issue 5, Pages 499-507
doi:10.1155/AAA.2005.499
On the range of the derivative of a smooth mapping between Banach spaces
Laboratoire Bordelais d'Analyse et Geométrie, Institut de Mathématiques de Bordeaux, Université de Bordeaux 1, 351 cours de la Libération, Talence Cedex 33405, France
Received 15 January 2004
Copyright © 2005 Robert Deville. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ℒ(X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f′(X)=A. Whenever f is only assumed Gâteaux differentiable, new phenomena appear: for instance,
there exists a mapping f from ℓ1(ℕ) into ℝ2, which is bounded, Lipschitz-continuous, and so that for all x,y∈ℓ1(ℕ), if x≠y, then ‖f′(x)−f′(y)‖>1.