J. Aust. Math. Soc.  75 (2003), 413-422
On -direct sums of Banach spaces and convexity

 Mikio Kato   Department of Mathematics   Kyushu Institute of Technology   Kitakyushu 804-8550   Japan  katom@tobata.isc.kyutech.ac.jp
 Kichi-Suke Saito   Department of Mathematics   Faculty of Science   Niigata University   Niigata 950-2181   Japan  saito@math.sc.niigata-u.ac.jp
 and
 Takayuki Tamura   Graduate School of Social Sciences and Humanities   Chiba University   Chiba 263-8522   Japan  tamura@le.chiba-u.ac.jp

Abstract
Let be Banach spaces and a continuous convex function with some appropriate conditions on a certain convex set in . Let be the direct sum of equipped with the norm associated with . We characterize the strict, uniform, and locally uniform convexity of by means of the convex function . As an application these convexities are characterized for the -sum (, ), which includes the well-known facts for the -sum in the case .