Research on geometric structures of Banach and function spaces with applications

我们得到了Banach空间几何结构的一系列结果，特别是关于Banach空间的n-直和的结果.主要结果如下。我们证明了von Neumann-Jordan常数和James常数之间的一个非常简单的不等式，它改进了以前关于它们之间关系的所有结果。我们还证明了另一个不等式，它给出了这些以前的结果的统一观点。对于n-直和，包括l_∞-和，我们刻画了一致非l ^n1性和弱近一致光滑性（对于n-多个Banach空间）.作为应用，给出了不动点性质和超自反性的一些结果。尖锐的平均三角形不等式，以及显示。

We obtained a sequence of results on geometric structures of Banach spaces, especially for their ψ-direct sums. Major results are as follows. We showed a quite simple inequality between the von Neumann-Jordan and James constants, which refined all the previous results on their relation. We also showed another inequality which gives a unifying sight on these previous results. For ψ-direct sums, including the l_∞-sum, we characterized the uniform non-l^n1-ness, and weak nearly uniform smoothness (for finitely many Banach spaces). As applications some results on the fixed point property and super-reflexivity were given. Sharp mean triangle inequality was shown as well.