On psi-direct sums of Banach spaces and convexity

J. Aust. Math. Soc. 75 (2003), 413-422

On $\psi$ -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 $X_1, X_2,\dots, X_N$ be Banach spaces and $\psi$ a continuous convex function with some appropriate conditions on a certain convex set in $\mathbb{R}^{N-1}$ . Let $(X_1\oplus X_2\oplus \cdots \oplus X_N)_{\psi}$ be the direct sum of $X_1,X_2,\dots,X_N$ equipped with the norm associated with $\psi$ . We characterize the strict, uniform, and locally uniform convexity of $(X_1\oplus X_2\oplus \cdots \oplus X_N)_{\psi}$ by means of the convex function $\psi$ . As an application these convexities are characterized for the $\ell_{p,q}$ -sum $(X_1\oplus X_2\oplus \cdots \oplus X_N)_{p,q}$ ( $1<q\leq p\leq \infty$ , $q<\infty$ ), which includes the well-known facts for the $\ell_p$ -sum $(X_1\oplus X_2\oplus \cdots \oplus X_N)_p$ in the case

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