Title
Approach groups
Author
Faculty/Department
Faculty of Sciences. Mathematics and Computer Science
Publication type
article
Publication
Provo, Utah ,
Subject
Mathematics
Source (journal)
The Rocky Mountain journal of mathematics. - Provo, Utah
Volume/pages
30(2000) :3 , p. 1057-1073
ISSN
0035-7596
ISI
000165670300018
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Affiliation
University of Antwerp
Abstract
Any normed vector space X is a topological group with respect to the norm topology and the underlying group operation of the vector space. Although for the majority of applications it is sufficient to knowthat this operation + : X ~ X ¨ X : (x, y) ¨ x + y is continuous, stronger properties of this mapping can be shown. In fact, if X ~X is equipped with the sum product metric, then addition becomes a contraction. Examples showth at different well-known topological (semi-)groups can be equipped with a natural metric (or gauge of metrics) such that addition is contractive. This approach group structure is a canonical generalization of topological groups (or metric groups in the sense of Parthasarathy) and shares some of the important features with the classical concept. For instance, every approach group allows for a natural uniformization.
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