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On the range of the derivatives of a smooth function between Banach spaces. (English) Zbl 1034.46039

The range of derivatives of a smooth function between Banach spaces is studied; the obtained results improve the main results from [D. Azagra and R. Deville, J. Funct. Anal. 180, 328–346 (2001; Zbl 0983.46016)] and generalize some results from [S. M. Bates, Isr. J. Math. 100, 209–220 (1997; Zbl 0898.46044)].
The authors provide conditions on a pair of Banach spaces \(X\) and \(Y\) which ensure the existence of a \(C^p\) smooth surjection \(f:X\to Y\) such that \(f\) vanishes outside a bounded set and the derivatives of \(f\) are all surjections. If \(X\) has a \(C^p\) smooth bump with bounded derivatives and \(\text{dens\,} X= \text{dens\,} {\mathcal L}^m_s(X,Y)\), then there exists another \(C^p\) smooth function \(f:X\to Y\) with bounded derivatives so that \(f\) vanishes outside the unit ball of \(X\) and \(f^{(k)}(X)\) contains the unit ball of \({\mathcal L}^m_s(X,Y)\) for all \(k=0,\dots,m\).
As an application, it is shown that every bounded starlike body of a separable Banach space \(X\) with a (Fréchet or Gateaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space \(X^*\). In the non-separable case, it is proved that \(X\) has such a property if \(X\) has a smooth partition of unity.

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
46B20 Geometry and structure of normed linear spaces
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