Approach groupsApproach groups
Faculty of Sciences. Mathematics and Computer Science

Fundamental Mathematics

article

2000Provo, Utah, 2000

Mathematics

The Rocky Mountain journal of mathematics. - Provo, Utah

30(2000):3, p. 1057-1073

0035-7596

000165670300018

E

English (eng)

University of Antwerp

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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