| Project/Area Number |
11640172
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | KYUSHU INSTITUTE OF TECHNOLOGY |
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Principal Investigator |
KATO Mikio KYUSHU INSTITUTE OF TECHNOLOGY, PROFESSOR, 工学部, 教授 (50090551)
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| Co-Investigator(Kenkyū-buntansha) |
HASHIMOTO Kazuo HIROSHIMA JOGAKUIN UNIVERSITY, PROFESSOR, 生活科学部, 教授 (00156275)
TAKAHASHI Yasuji OKAYAMA PREFECTURAL UNIVERSITY, PROFESSOR, 情報工学部, 教授 (30001853)
KOBAYASHI Takayuki KYUSHU INSTITUTE OF TECHNOLOGY, ASSOCIATE PROFESSOR, 工学部, 助教授 (50272133)
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| Project Period (FY) |
1999 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| Budget Amount *help |
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2000: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 1999: ¥1,900,000 (Direct Cost: ¥1,900,000)
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| Keywords | von Neumann-Jordan constant / James constant / uniform normal structure / Clarkson-type inequality / Rademacher type-cotype / absolute norm / direct sum of Banach spaces / Geometry of Banach spaces / convex function / uniformly convex space / Jordan-von Neumann constant / absolute norm on C^n / Clarkson-type inequality / type,cotype / Banach space / Geometry of Banach spaces / normal structure / interpolation |
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| Research Abstract |
Geometric properties of Banach spaces, as well as their related norm inequalities, are investigated from an operator theoretical point of view. This approach enables us to apply interpolation techniques to research on the Banach space geometry. Major results are as follows.
1. On Clarkson-type inequalities and Rademacher type-cotype :
(1) A sequence of multi-dimensional Clarkson-type inequalities, generalized Clarkson, random Clarkson inequalities and thier variants, are characterized in terms of Rademacher type and cotype. These inequalities are equivalent in a Lebesgue-Bochner space.
(2) We extended q-uniform convexity and p-uniform smoothness inequalities in parameters and in number of elements.
2. On geometric constants of Banach spaces :
(1) We clarified some relations between the von Neumann-Jordan (NJ-) constant and James constant, resp., the normal structure coefficient. In particular, if X has the NJ-constant less than 5/4, then X, as well as the dual space X^*, has the uniform normal structure and hence the fixed point property. An answer was also presented to the question of Gao and Lau concerning James constant.
(2) We determined the NJ-and James constants of 2-dimensional Lorentz sequence spaces d (w, q).
(3) The supremum of p, 1【less than or equal】p【less than or equal】, for which a subspace X of L_1 is that of L_p was determined by the NJ-constant of X.
3. Out absolute norms :
(1) The NJ-constant of absolute normalized norms on C^2 was determined and estimated. Also we showed that all these norms are uniformly non-square except the l_1-and l_∞-norms.
(2) The correspondence between the absolute normalized norms on C^2 and the convex functions ψon [0, 1] with certain conditions was extended to the n-dimensional case.
(3) By using absolute norms we introduced the notion of ψ-direct sum of a finite number of Banach spaces, and extended the well-known facts for the l_p-sums of Banach spaces concerning strict resp. uniform convexity.
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