报告人  ：Mikio KATO教授

          Department of Mathematics, Shinshu Univ.

报告题目：On Geometry of Banach spaces I, II.An introduction and some recent results

报告时间：2016年02月26日 下午15:00

报告地点：海韵数理楼661

学院联系人：张文副教授

报告摘要：In the first half part an introduction to “Geometry of Banach spaces” will be presented. It will start with some basic notions such as strict convexity, uniform convexity, and uniform non-suquareness, etc., where we shall see how these notions appear naturally in the context with well-known approximation problems.

Secondly in this part, the “Sharp Triangle Inequality” will be introduced, which was given by the present author, etc. in Math. Inequal. Appl., 2007. This is often powerful in proofs and their simplification. Some applications will be given.

 Thirdly, we shall discuss geometric constants, especially, modulus of convexity, von Neumann-Jordan constant, James constant and their modifications. As is well known, Lp-spaces (1 < p < ∞) are uniformly convex, while L2 is “more uniformly convex” than Lp (p ̸= 2). Thus the notion of uniform convexity cannot distinguish this difference. Generally speaking, “geometric constants” enables us to know such a “quantative”information on various geometric properties! This is their importance.

In the latter half of this talk, we shall discuss direct sums of Banach spaces. In particular, the notion of ψ-direct sum will be introduced, which has advantage in the point that we can use the convex function ψ, a powerful tool, to investigate various properties of direct sums, especially to construct examples. A sequence of recent results will be presented concerning uniform non-squareness and more generally uniform non-ℓn1 -ness.

 欢迎广大师生参加！