RESEARCH ON GEOMETRY OF BANACH AND FUNCTION SPACES AND INEQUALITIES

研究了Banach空间和函数空间的几何结构，特别是与某些范数不等式有关的几何结构。我们还研究了Banach空间的ψ直和，这似乎很重要，因为我们可以很容易地从凸函数ψ构造出许多具有非L p型范数的Banach空间的例子。主要结果如下：[1]关于Banach空间、函数空间的几何结构和范数不等式：(1)我们推广了强随机Clarkson不等式，刻画了所得到的不等式成立的Banach空间是p-一致光滑的，或者等价地刻画了具有相同常数的强型p-一致光滑的Banach空间。(2)引入了带权的Hanner型不等式，并用这些不等式刻画了2-一致光滑和2-一致凸的空间。通过改变权的位置，得到的不等式刻画了p一致光滑和q一致凸的空间。确定了L p-空间中这些不等式中的权值的最佳值。[3]关于Banach空间的ψ-直和：(1)刻画了Banach空间的ψ-直和的光滑性、弱近一致光滑性、价值性质、Schur性等性质.作为应用，我们给出了几个非一致非方的Banach空间的例子，(2)刻画了Banach空间的一致非L n_1-凸性(B_n-凸性)，特别是刻画了空间的L_1-和、L_∞-和，得到了一些关于James常数、性质M与不动点性质的关系、关于Banach-Mazur距离与超自反性的关系等结果。

Geometric structures of Banach and function spaces were investigated especially in relation with some norm inequalities. Also ψ-direct sums of Banach spaces were investigated, which seem important as we can easily construct many examples of Banach spaces with a non l_p-type norm from a convex function ψ. Major results are as follows.[1]On geometric structures of Banach and function spaces and norm inequalities :(1)We extended the strong random Clarkson inequality, and characterized the Banach spaces in which the resulting inequality is valid are those of p-uniform smooth, or equivalently those of strong type p with the same constants involved them.(2)We introduced Hanner-type inequalities with a weight and characterized 2-uniformly smooth, and 2-uniformly convex spaces with these inequalities. By changing the place of the weight, the resulting inequalities characterize p-uniformly smooth, and q-uniformly convex spaces. The best value of the weight in these inequalities were determined for L_p-spaces. A duality theorem for these Hanner-type inequalities was presented as well.[2]We presented a sharp triangle inequality and its reverse in a Banach space.[3]On ψ-direct sums of Banach spaces :(1)We characterized the following properties of ψ-direct sums of Banach spaces :smoothness, weak nearly uniform smoothness, WORTH property, Schur property etc. As an application we presented several examples of Banach spaces which are not uniformly non-square, but have the fixed point property.(2)We characterized uniform non l^n_1-ness (B_n-convexity) particularly for the l_1-sum and l_∞-sum of Banach spaces.[4]Some resuts were obatined on the James constant, on the relation between Property M and the fixed point property, and on the Banach-Mazur distance in relation with super-reflexivity etc.