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International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 80846, 9 pages
http://dx.doi.org/10.1155/IJMMS/2006/80846

-convexity in infinite-dimensional Banach spaces and applications to Kergin interpolation

Division of Mathematics, Royal Institute of Technology (KTH), Stockholm 100 44, Sweden

Received 2 February 2005; Revised 7 April 2006; Accepted 25 April 2006

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the concepts of linear convexity and -convexity in complex Banach spaces. The main result is that any -convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given a -convex domain Ω in the Banach space X and a point pΩ, there is a complex hyperplane through p that does not intersect Ω. We also prove that linearly convex domains are holomorphically convex, and that Kergin interpolation can be performed on holomorphic mappings defined in -convex domains.