On the range of the derivative of a real-valued function with bounded support

• Autorzy: T. Gaspari
• Języki publikacji: EN

Abstrakty

EN

We study the set f'(X) = {f'(x): x ∈ X} when f:X → ℝ is a differentiable bump. We first prove that for any C²-smooth bump f: ℝ² → ℝ the range of the derivative of f must be the closure of its interior. Next we show that if X is an infinite-dimensional separable Banach space with a $C^{p}$-smooth bump b:X → ℝ such that $||b^{(p)}||_{∞}$ is finite, then any connected open subset of X* containing 0 is the range of the derivative of a $C^{p}$-smooth bump. We also study the finite-dimensional case which is quite different. Finally, we show that in infinite-dimensional separable smooth Banach spaces, every analytic subset of X* which satisfies a natural linkage condition is the range of the derivative of a C¹-smooth bump. We then find an analogue of this condition in the finite-dimensional case

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 46T20: Continuous and differentiable maps
• 26B05: Continuity and differentiation questions
• 46G05: Derivatives

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2002
• Tom: 153
• Numer: 1
• Strony: 81-99

Daty

wydano

2002

Twórcy

autor

T. Gaspari

• Mathématiques Pures de Bordeaux (MPB), UMR 5467 CNRS, Université Bordeaux 1, 351, cours de la Libération, 33400 Talence, France

Identyfikatory

DOI

10.4064/sm153-1-6