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\(\mathbb C\)-convexity in infinite-dimensional Banach spaces and applications to Kergin interpolation. (English) Zbl 1135.46007

An open subset \(\Omega\) of a complex Banach space is said to be \({\mathbb C}\)-convex if \(\Omega \cap \ell\) is a simply connected subset of \(\ell\) for each affine complex line \(\ell\), and it is called linearly convex if for each \(x \in X\setminus \Omega\) there is an affine complex hyperplane \(H\) containing \(x\) and not intersecting \(\Omega\). It is easy to see that ordinary convexity implies \({\mathbb C}\)-convexity, and as the main result of this paper it is shown that \({\mathbb C}\)-convexity implies linear convexity in all complex Banach spaces. This can be considered as a complex version of the Hahn-Banach or Mazur Theorem. Finally, it is shown that the Kergin interpolation which was constructed originally for \(C^k\)-mappings on open convex subsets \(U\) of a Banach space can also be constructed for holomorphic mappings on open \({\mathbb C}\)-convex sets.

MSC:

46B20 Geometry and structure of normed linear spaces
46G25 (Spaces of) multilinear mappings, polynomials
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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References:

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