| 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)
小林 孝行 九州工業大学, 工学部, 助教授 (50272133)
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| Budget Amount *help |
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2003: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2002: ¥1,800,000 (Direct Cost: ¥1,800,000)
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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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