International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 80846, 9 pages
doi:10.1155/IJMMS/2006/80846
ℂ-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 Lars Filipsson. 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 ℂ-convexity
in complex Banach spaces. The main result is that any
ℂ-convex domain is necessarily linearly convex. This is a
complex version of the Hahn-Banach theorem, since it means the
following: given a ℂ-convex domain Ω
in the Banach space
X
and a point p∉Ω, 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 ℂ-convex domains.