Normed versus topological groups: dichotomy and duality

The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics [BO-TI], and embracing its classical univariate counterpart (cf. [BGT]) as well as fragmentary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]), are groups with a right-invariant metric carrying flows. Following the vector paradigm, they are best seen as normed groups. That concept only occasionally appears explicitly in the literature despite its frequent disguised presence, and despite a respectable lineage traceable back to the Pettis closed-graph theorem, to the Birkhoff-Kakutani metrization theorem and further back still to Banach's Theorie des operations lineaires. Its most recent noteworthy appearance has been in connection with the Effros Open Mapping Principle. We collect together known salient features and develop their theory including Steinhaus theory unified by the Category Embedding Theorem [BO-LBII], the associated themes of subadditivity and convexity, and a topological duality inherent to topological dynamics. We study the latter both for its independent interest and as a foundation for topological regular variation.