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On Ψ direct sums of Banach spaces and convexity

Part of: Real functions General convexity Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

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

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Let X1, X2, …, XN be Banach spaces and ψ a continuous convex function with some appropriate conditions on a certain convex set in RN−1. Let (X1⊕X2⊕…⊕XN)Ψ be the direct sum of X1, X2, …, XN equipped with the norm associated with Ψ. We characterize the strict, uniform, and locally uniform convexity of (X1 ⊕ X2 ⊕ … ⊕ XN)Ψ; by means of the convex function Ψ. As an application these convexities are characterized for the ℓp, q-sum (X1 ⊕ X2 ⊕ … ⊕ XN)p, q (1 < q ≤ p ≤ ∈, q < ∞), which includes the well-known facts for the ℓp-sum (X1 ⊕ X2 ⊕ … ⊕ XN)p in the case p = q.

Keywords

absolute normconvex functiondirect sum of Banach spacesstrictly convex spaceuniformly convex spacelocally uniformly convex space

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 46B99: None of the above, but in this section 26A51: Convexity, generalizations 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.) 90C25: Convex programming
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 3 , December 2003 , pp. 413 - 422
Copyright
Copyright © Australian Mathematical Society 2003

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