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This is a second attempt after that my previous post on the same subject has been closed since, I was said, "not a real question". Pas de problème ! I hope that it is now a little bit clearer than before.

It is apparent that metric and normed (vector) spaces have profound structural differences. Loosely speaking, a metric space is a "bare" set $X$ together with a function $d: X \times X \to X$ satisfying some prescribed properties, while a normed vector space is an algebraic structure $V$ along with a function $n: |V| \to |V|$ fulfilling a few other axioms (I'm using $|V|$ for the carrier of $V$): The former is called a distance, the latter a norm. But names do not always reflect the essential nature of concepts and they happen to mark differences that can be partly smoothed out provided that one can move uphill and watch the valley from the mountain peak.

In this line of thought, we can all agree, I think, that some (basic) issues, results and constructions local to the theory of metric spaces are conceptually close to (basic) issues, constructions and results local to the theory of normed (vector) spaces: This is the case, e.g., with the most usual choices of the class of morphisms to associate with the categories $\bf Met$ (metric spaces with Lipschitz maps as arrows) and $\bf Nor$ (normed spaces with bounded linear transformations as arrows) and with their "short variants"; with binary products and coproducts (and indeed with arbitrary nonempty products and coproducts if we let metrics and norms be valued in $[0,+\infty]$ rather than in $\mathbb{R}_0^+$); or even with some of the conditions required, with the notation from above, by the definition itself of $d$ and $n$ (I'm especially thinking of the triangle inequality). These similarities are all the more striking, I think, if we enlarge our scope and bring other normed structures into the picture (e.g. normed groups [1], valuated rings [2], and normed algebras): Let me refer to these, in what follows, as normed algebraic structures (it is not that important, I think, to know what it is formally meant by a normed algebraic structure). One is then naturally prompted to raise the question: Does there exist some higher perspective (say, some sort of abstract theoretical framework, hopelessly shaped on the language of categories), where algebraic normed structures are all special instances of one same archetypical structure? But this is not my question. Instead, my question is

whether or not anything similar has been already done in the literature in relation to the construction of a unified framework where metric spaces AND algebraic normed structures can be all regarded as instances of the one same archetypical structure.

Here, the term "structure" must be formally intended as "model of a finitary first-order theory (in one or many sorts) interpreted over $\bf Set$ or another category $\bf C$ with some appropriate properties" (the underlying formal language, say, is that of the NBG class theory).

Notes. [1] See, e.g., N.H. Bingham and A.J. Ostaszewski, Normed groups: dichotomy and duality, CDAM Research Report Series, LSE-CDAM-2008-10, London School of Economics, 2008 and references therein. [2] I'm using the term "valuation" as a synonymous of "absolute value".

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closed as not a real question by Yemon Choi, quid, Suvrit, Deane Yang, Salvo Tringali May 26 '12 at 19:08

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Things with norms have a distinguished point, namely the element of norm zero – Yemon Choi May 26 '12 at 14:17
Name me an example of something that is commonly called a normed algebraic structure which does not possess a distinguished point, namely the unique element of norm zero. – Yemon Choi May 26 '12 at 15:35
A linear normed space is naturally a metric space. Metric spaces are a much wider class. In fact, up to isometries, they are all subsets of metric spaces. That said, I can't really see how one could see norms and metrics as "to sides of the same coin" – Pietro Majer May 26 '12 at 16:15
I also vote to close this instance of the question. I do not understand what you are looking for. For a metric one has some sort of distance and for a norm some sort of size that typically can be turned into a distance by considering the size of the difference (cf also Yemon's comment). And now you have been told twice that you can 'realize' any metric space as some subset of a normed space. This seems enough of an answer to me. – quid May 26 '12 at 17:32
Meta thread:… – Yemon Choi May 26 '12 at 18:11