| Co-Investigator(Kenkyū-buntansha) |
SAITO Kichi-suke NIIGATA UNIVERSITY, Professor, 理学部, 教授 (30018949)
TAKAHASHI Yasuji OKAYAMA PREFECTURAL UNIVERSITY, Professor, 情報工学部, 教授 (30001853)
OKAZAKI Yoshiaki Kyushu Institute Of Technology, Professor, 情報工学部, 教授 (40037297)
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| Research Abstract |
Geometric structures of Banach and function spaces are investigated, especially in relation with the notions of Rademacher type and cotype. Also φ-direct sums of Banach spaces are investigated, which seem important as one can easily, construct a plenty of examples of Banach spaces with a non ι_ptype norm from a convex function φ. Major results are as follows.
1.On geometric structures and type, cotype :
(1) We introduced the notions of strong (Rademacher) type and cotype, and characterized ρ-uniformly smooth, and q-uniformly convex spaces with these properties. The heredity of these properties to Lebesgue-Bochner spaces L_γ(X) was shown, as well.
(2) We introduced strong random Clarkson inequality, and proved that this inequality holds in a Banach space if and only if the space is of strong type ρ, or equivalently, ρ-uniformly smooth. (Recall that random Clarkson inequality holds in a Banach space if and only if the space has type ρ ; Kato-Persson-Takahashi, Collect. Math. 51 (2000).)
2. On geometric structures and norm inequalities :
(1) Hanner-type inequalities are often useful to treat properties described with the modulus of convexity of a Banach space. We considered Hanner-type inequalities, especially, those with a weight and their n element version, and characterized optimal 2-uniform convexity and uniform non-squareness, etc. with these inequalitites :
(2) We introduced a Schaffer-type constant for a Banach space and showed a relation with uniform normal structure.
3. On φ-direct sums of Banach spaces :
(1) We characterized the following properties of a φ-direct sum of Banach spaces : strict, uniform convexity, uniform non-squareness, uniform non Ι^n_1-ness, reflexivity, weak nearly uniform smoothness, smoothness, etc.
(2) The James constant of an absolute norm on R2 was calculated for some cases, which gives a partial answer to a problem of Kato-Maligranda (JMAA 258 (2001)).
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