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New Results on the Aggregation of Norms
Mathematics ( IF 2.4 ) Pub Date : 2021-09-17 , DOI: 10.3390/math9182291
Tatiana Pedraza , Jesús Rodríguez-López

It is a natural question if a Cartesian product of objects produces an object of the same type. For example, it is well known that a countable Cartesian product of metrizable topological spaces is metrizable. Related to this question, Borsík and Doboš characterized those functions that allow obtaining a metric in the Cartesian product of metric spaces by means of the aggregation of the metrics of each factor space. This question was also studied for norms by Herburt and Moszyńska. This aggregation procedure can be modified in order to construct a metric or a norm on a certain set by means of a family of metrics or norms, respectively. In this paper, we characterize the functions that allow merging an arbitrary collection of (asymmetric) norms defined over a vector space into a single norm (aggregation on sets). We see that these functions are different from those that allow the construction of a norm in a Cartesian product (aggregation on products). Moreover, we study a related topological problem that was considered in the context of metric spaces by Borsík and Doboš. Concretely, we analyze under which conditions the aggregated norm is compatible with the product topology or the supremum topology in each case.

中文翻译:

规范聚合的新结果

如果对象的笛卡尔积产生相同类型的对象,这是一个自然的问题。例如,众所周知,可度量拓扑空间的可数笛卡尔积是可度量的。与这个问题相关,Borsík 和 Doboš 描述了那些允许通过聚合每个因子空间的度量在度量空间的笛卡尔积中获得度量的函数。Herburt 和 Moszyńska 也研究了这个问题的规范。可以修改该聚合过程,以便分别通过一系列度量或规范在某个集合上构建度量或规范。在本文中,我们描述了允许将向量空间上定义的(非对称)范数的任意集合合并为单个范数(集合上的聚合)的函数)。我们看到这些函数不同于那些允许在笛卡尔积(积聚合)中构建范数的函数。此外,我们研究了 Borsík 和 Doboš 在度量空间背景下考虑的相关拓扑问题。具体来说,我们分析了聚合范数在每种情况下与乘积拓扑或最高拓扑兼容的条件。
更新日期:2021-09-17
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