On the Range of the Derivative of a Smooth Mapping between Banach Spaces
نویسنده
چکیده
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 subsetA of (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 1(N) intoR2, which is bounded, Lipschitz-continuous, and so that for all x, y ∈ 1(N), if x = y, then ‖ f ′(x)− f ′(y)‖ > 1.
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