New Results on the Aggregation of Norms

Tatiana Pedraza and Jesús Rodríguez-López
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Tatiana Pedraza: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
Jesús Rodríguez-López: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain

Mathematics, 2021, vol. 9, issue 18, 1-19

Abstract: 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 Moszy?ska. 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.

Keywords: norm; asymmetric norm; aggregation; product topology; supremum topology (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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