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

自James Clarkson于1936年引入一致凸性的概念以来，对Banach空间的凸性和光滑性的研究形成了泛函分析的一个重要分支。从理论和应用的角度来看，这些性质都很有趣。巴拿赫空间理论的一个主要问题是关于泛函分析的典型结构的这些几何性质的稳定性，特别是包括直接和和向量值函数的空间，如Orlicz-Bochner空间或更一般的Köthe-Bochner空间。关于这类空间的经典凸性和光滑性已经有大量的文献。在我的论文中，我对直接和和Köthe-Bochner空间中的一些广义的凸性和平滑性（所谓的acs性质）进行了广泛的研究。直接和和Köthe-Bochner空间的概念更广泛的推广是所谓的巴拿赫空间的直接积分，1991年由Haydon， Levy和Raynaud引入，用于研究所谓的随机巴拿赫空间的表示。到目前为止，这些直接积分的几何性质还没有被系统地研究过。因此，对直接积分的凸性和光滑性进行全面的研究是本课题的主要目的。这也需要对直接积分的对偶空间的显式描述，这是一个独立的兴趣。这样的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Köthe-Bochner空间的已知对偶理论的基础上。计划研究经典性质（如严格凸性或一致凸性，范数的Gateaux或frechet可微性）以及上述acs性质和其他几个关于直积分凸性的广义概念。新的结果，特别是在几何无穷直接和和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.