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A $\mathcal {C}^k$-seeley-extension-theorem for Bastiani’s differential calculus

Part of: Maps and general types of spaces defined by maps Calculus on manifolds; nonlinear operators Measures, integration, derivative, holomorphy

Published online by Cambridge University Press:  20 October 2021

Maximilian Hanusch Open the ORCID record for Maximilian Hanusch [Opens in a new window]
Maximilian Hanusch*
Affiliation:
Institut für Mathematik, Universität Paderborn, Paderborn, Germany
*
e-mail: mhanusch@math.upb.de
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Abstract

We generalize a classical extension result by Seeley in the context of Bastiani’s differential calculus to infinite dimensions. The construction follows Seeley’s original approach, but is significantly more involved as not only $C^k$-maps (for ) on (subsets of) half spaces are extended, but also continuous extensions of their differentials to some given piece of boundary of the domains under consideration. A further feature of the generalization is that we construct families of extension operators (instead of only one single extension operator) that fulfill certain compatibility (and continuity) conditions. Various applications are discussed as well.

Keywords

Extension of functions in infinite-dimensional spacesSeeley’s extension theoremBastiani’s differential calculus

MSC classification

Primary: 46G05: Derivatives
Secondary: 54C20: Extension of maps 58C20: Differentiation theory (Gateaux, Fréchet, etc.)

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 1 , February 2023 , pp. 170 - 201
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Copyright
© Canadian Mathematical Society 2021

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