【作者】 乌日娜；

【导师】 苏雅拉图；

【作者基本信息】 内蒙古师范大学 ， 基础数学， 2013， 硕士

【摘要】 Banach空间的几何性质（如凸性和光滑性）是Banach空间理论中的重要研究内容之一，其研究的内容不仅是泛函分析的重要内容之一,也是当今数学极具理论意义和应用价值的国际前沿性研究课题。它的建立和发展不仅扩展了泛函分析学科的内容,而且为其他学科和技术领域带来了更为广泛的应用。随着Banach空间几何理论的迅速发展，作为它的重要内容之一---凸性与光滑性也得到了很好的研究。凸性与光滑性的研究在Banach空间几何理论的研究中占据着重要的地位,而合理引进并讨论某种凸性(光滑性)以及与其具有对偶关系的光滑性(凸性)或发现严格介于某两种凸性(光滑性)之间的另一种凸性(光滑性)，对于揭示凸性(光滑性)的本质、比较不同凸性的强弱程度(相应地,比较不同光滑性的好坏程度)、建立凸性与光滑性之间的对偶理论有重要意义。本文研究了Banach空间的某些凸性与光滑性,得到了较好的结果,全文共分为三章。第一章：讨论了局部一致光滑,局部完全k光滑,局部一致光滑,一致极光滑和(WM)性质的关系,发现了这些光滑性之间的内在联系,完善了以前关于这些光滑空间的有关结论。第二章：研究了一致极凸空间的推广---k一致极凸空间,并给出了其特征刻画以及与其它凸性之间的关系。证明了这一类新的Banach空间严格介于k一致凸空间和强凸空间之间，也严格介于k-完全凸空间和k-强凸空间之间,但它与局部一致极凸空间之间不存在蕴含关系。第三章：对Banach空间X的共轭空间X~*引入了涉及不同拓扑的k可凹点,利用Banach空间几何技巧证得:是光滑空间当且仅当S (X~*)的范数可达到点是U (X~*)的第二类W~*k可凹点；是非常光滑空间当且仅当的范数可达到点是的W-k可凹点；是强光滑空间当且仅当的范数可达到点是的第一类W~*-k可凹点。此外，给出了第一类W~*-k可凹点和切片之间的关系。更多还原

【Abstract】 In this thesis, some convexity and smoothness are exploredmainly, and some productive results are obtained in the process ofmy study.This thesis consists of three parts.Chapter one: In this paper, the relationship among the locallyuniformly smoothness, locally fully k-smoothness, locally k-uniformlysmoothness, uniformly extremely smoothness and property (WM) arestudied and have been show completely. The previous results aboutsuch smoothness are completed.Chapter two: A new class of Banach spaces which are the ge-neralizetion of the uniformly extremely convex spaces which wereintroduced by Suyalatu and Wudunqiqige are introduced. It is proved thatthe new class of Banach spaces lies strictly between either the classes ofk-uniformly rotund spaces and k-strongly convex spaces or classes offully k-convex spaces and k-strongly convex spaces, and has no inclusiverelation with the class of locally k-uniformly convex uniformly spaces. Inaddition, some characterizations and properties of this new class ofBanach spaces are obtained. In particular, the results of Suyalatu andWudunqiqige are included in these results which are obtained in thischapter.Chapter three: In this paper, the k denting points respect to weaktopology (respectively, weak*topology, norm topology) ofX~*areintroduced. It is proved that (i) X is k-smooth spaces if and only if the points of S(X~*)which attains its norm are the second type W*-k dentingpoints of U(X~*).(ii) X is k-very smooth spaces if and only if the pointsof which attains its norm are the W-k denting points of.(iii)is k-strongly smooth spaces if and only if the points of whichattains its norm are the first type W*-k denting points of. Also, therelations between first type W~*-k denting points and slice of aregiven.更多还原