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

Robert Deville

Abstract and Applied Analysis, 2005, vol. 2005, 1-9

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 ℓ 1 ( ℕ ) into ℝ 2 , which is bounded, Lipschitz-continuous, and so that for all x , y ∈ ℓ 1 ( ℕ ) , if x ≠ y , then ‖ f ′ ( x ) − f ′ ( y ) ‖ > 1 .

Date: 2005
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:831516

DOI: 10.1155/AAA.2005.499

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