论文标题
关于哈达姆空间及其应用的较弱拓扑
On a weak topology for Hadamard spaces and its applications
论文作者
论文摘要
我们研究了哈达玛空间中现有的弱顺序收敛概念是否可以由拓扑引起。我们提供了一个我们所谓的弱适当的Hadamard空间的答案。提出了双重空间的概念，这表明我们的弱拓扑和双空间与希尔伯特空间的标准空间一致。此外，我们介绍了地球段和相应的弱拓扑空间，我们表明该空间对其底层Hadamard空间是同质的。作为它的应用，我们表明了一个地球段的存在，该段充当了地球上可区分函数的最陡下降方向，该函数的大地衍生物满足了某些特性。最后，我们将几个结果从经典的功能分析扩展到Hadamard空间的设置，并将拓扑与其他现有的较弱拓扑概念进行了比较。
We investigate if an existing notion of weak sequential convergence in a Hadamard space can be induced by a topology. We provide an answer on what we call weakly proper Hadamard spaces. A notion of dual space is proposed and it is shown that our weak topology and dual space coincide with the standard ones in the case of a Hilbert space. Moreover we introduce the space of geodesic segments and a corresponding weak topology, and we show that this space is homeomorphic to its underlying Hadamard space. As an application of it we show the existence of a geodesic segment that acts as direction of steepest descent for a geodesically differentiable function whose geodesic derivative satisfies certain properties. Finally we extend several results from classical functional analysis to the setting of Hadamard spaces, and we compare our topology with other existing notions of weak topologies.
