On the range of the derivative of a smooth mapping between Banach spaces

Robert Deville

Abstract and Applied Analysis, 2005, vol. 2005, issue 5, 499-507

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.

Date: 2005
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https://doi.org/10.1155/AAA.2005.499

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2005:y:2005:i:5:p:499-507

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