Every metric is a norm [closed]

Hi there.

Normed structures, and notably normed groups, valuated rings, normed (vector) spaces (over a valuated field), and normed algebras among the others, are familiarly considered as sort of special metric spaces. Thus, according to the common standpoint, one would say that "metrics are on a higher step than norms on the hierarchy ladder of abstraction" (basically for the fact that we have a way to equip them with a "seemingly natural" metric but no "seemingly natural" way to think of an arbitrary metric space as a normed structure), and I was actually taught to think of it in this mood. But now, I'm really wondering if there exists previous work providing a (completely) different outlook, or at least suggesting that the "truth" may be somewhere else in the other view. Of course, I've already looked through Lawvere's well-known paper on metric spaces and generalized logic, but no hints seem to be there to the picture that I'm trying to draw. Then, my question is:

Are you aware of any previous literature pursuing the idea that metrics can be ultimately regarded as a special kind of "norms", to the degree that metric spaces can be recovered as an instance of abstract normed structures of some appropriate type (where the word type has, say, the formal meaning of model theory)?

I've no clue how to tag this question. I was thinking of "total temporary insanity", but I couldn't find it in the list.

P.S.: this is an edit following up the comments below (where the interested reader can find some references to the ubiquitous presence of norms beyond the scope of the theory of Banach spaces).

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I don't see any real question here. Norms are only defined on vector spaces, and not all metrics on vector spaces come from norms. I suppose you wish to make some heuristics in the opposite direction. You could try to formulate a precise question at least related to your goals so that a wider range of people could try to help. – Fernando Muro May 26 2012 at 8:44
Evvery metric space is isometric to a subset of a normed vector space. This is both standard and easy. – Laurent Berger May 26 2012 at 8:52
@Fernando. How do you happen to say that norms are only defined on vector spaces? M. Grandis has, e.g., a notion of norm for some additive categories, with applications in algebraic topology (see Categories, norms and weights, Journal of Homotopy and Related Structures, Vol. 2, No. 2, pp. 171-186, 2007). This has been further worked out by G.S.H. Cruttwell in his 2008 PhD thesis to study an analogue of the change of basis for enriched categories. Also, group norms have been studied for decades in relation to the geometrical theory of groups and Gromov's work in the subject. (tbc) – Salvo Tringali May 26 2012 at 9:18
Well, call me ignorant, but those are the norms I know, and it is the classical notion of norm I'd say most people know: en.wikipedia.org/wiki/Norm_(mathematics) If you wanted to consider some generalizations you could (actually, I think you should) have given some details within your question. – Fernando Muro May 26 2012 at 9:39
Vote to close as not a real question; with all the discussionn what is or is not a 'norm' I cannot see where this should be going. – quid May 26 2012 at 11:26