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On smooth extensions of vector-valued functions defined on closed subsets of Banach spaces

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Abstract

Let \(X\) and \( Z\) be Banach spaces, \(A\) a closed subset of \(X\) and a mapping \(f:A\rightarrow Z\). We give necessary and sufficient conditions to obtain a \(C^1\) smooth mapping \(F:X \rightarrow Z\) such that \(F_{\mid _A}=f\), when either (i) \(X\) and \(Z\) are Hilbert spaces and \(X\) is separable, or (ii) \(X^*\) is separable and \(Z\) is an absolute Lipschitz retract, or (iii) \(X=L_2\) and \(Z=L_p\) with \(1<p<2\), or (iv) \(X=L_p\) and \(Z=L_2\) with \(2<p<\infty \), where \(L_p\) is any separable Banach space \(L_p(S,\Sigma ,\mu )\) with \((S,\Sigma ,\mu )\) a \(\sigma \)-finite measure space.

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Acknowledgments

The authors wish to thank the reviewer for the suggestions and comments. L. Sánchez-González conducted part of this research while at the Institut de Mathématiques de Bordeaux. This author is indebted to the members of the Institut and very especially to Robert Deville for their kind hospitality and for many useful discussions. The authors wish to thank Jesús M.F. Castillo, Ricardo García and Jesús Suárez for several valuable discussions concerning the results of this paper.

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  1. Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040, Madrid, Spain

    M. Jiménez-Sevilla & L. Sánchez-González

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  1. M. Jiménez-Sevilla
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  2. L. Sánchez-González
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Correspondence to L. Sánchez-González.

Additional information

Supported in part by DGES (Spain) Project MTM2009-07848. L. Sánchez-González has also been supported by Grant MEC AP2007-00868.

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Jiménez-Sevilla, M., Sánchez-González, L. On smooth extensions of vector-valued functions defined on closed subsets of Banach spaces. Math. Ann. 355, 1201–1219 (2013). https://doi.org/10.1007/s00208-012-0814-0

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