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On the range of the derivative of a smooth mapping between Banach spaces. (English) Zbl 1106.46023

Summary: 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 \({\mathcal L}(X,Y)\) 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 \(f:\ell^1(\mathbb{N})\to\mathbb{R}^2\) which is bounded, Lipschitz-continuous, and so that for all \(x,y\in\ell^1(\mathbb{N})\), if \(x\neq y\), then \(\|f'(x)-f'(y)\|>1\).

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
46T20 Continuous and differentiable maps in nonlinear functional analysis
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