Geometric properties in Orlicz spaces, direct sums and Banach spaces of vector-valued functions

自从1936年James克拉克森引入一致凸性的概念以来，Banach空间的凸性和光滑性的研究成为泛函分析的一个重要分支。从理论和应用的角度来看，这些性质都是有趣的。Banach空间理论的一个主要问题涉及这些几何性质相对于泛函分析的典型构造的稳定性，其中包括特别是直接和空间的向量值的功能，如Orlicz-Bochner空间或更一般的Köthe-已经有大量的文献讨论了这种空间的经典凸性和光滑性。在我的论文中，我对一些广义的凸性和光滑性概念进行了广泛的研究（所谓的acs性质）。直接和Köthe-Bochner空间的概念的更广泛的推广是所谓的Banach空间的直接积分，它是由Haydon在1991年引入的，Levy和Raynaud研究了所谓的随机Banach空间的表示。到目前为止，这些直接积分的几何还没有系统的研究，因此，本项目的主要目的是对直接积分的凸性和光滑性进行全面的研究。这也需要一个明确的描述的对偶空间的直接积分，这是独立的利益。这种描述直到现在还不为人所知，因此它必须在这个项目中发展，建立在已知的Köthe-Bochner空间的对偶理论上。G.严格或一致凸性，规范的Gateaux-或Frechet-可微性）以及上述acs性质和直接积分中凸性的其他几个广义概念。新的结果，特别是几何的无穷直和和Orlicz/Köthe-Bochner空间也有望。

Since James Clarkson introduced the notion of uniform convexity in 1936, the study of convexity and smoothness properties of Banach spaces forms an important subfield of functional analysis. Such properties are interesting both from a theoretical and a more applied point of view.A principal question of Banach space theory concerns the stability of these geometric properties with respect to typical constructions of functional analysis, which include in particular direct sums and spaces of vector-valued functions such as Orlicz-Bochner spaces or the more general Köthe-Bochner spaces.There exists already a wide literature on classical convexity and smoothness properties of such spaces. In my dissertation, I have conducted extensive studies on some generalised notions of convexity and smoothness (the so called acs properties) in direct sums and Köthe-Bochner spaces.A far more wide-reaching generalisation of both the concepts of direct sums and Köthe-Bochner spaces are the so called direct integrals of Banach spaces, which were introduced in 1991 by Haydon, Levy and Raynaud for the study of representations of so called random Banach spaces. So far, the geometry of these direct integrals has not been studied systematically.It is therefore the main aim of this project to conduct comprehensive research on convexity and smoothness properties of direct integrals. This requires also an explicit description of the dual space of a direct integral, which is of independent interest. Such a description is not known until now and thus it has to be developed within this project, building on the known duality theory for Köthe-Bochner spaces.It is planned to study classical properties (e. g. strict or uniform convexity, Gateaux- or Frechet-differentiability of the norm) as well as the above-mentioned acs properties and several other generalised notions of convexity in direct integrals. New results especially on the geometry of infinte direct sums and Orlicz-/Köthe-Bochner spaces are also expected.