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Some time ago, I was thinking about whether it would be possible to generalize some results from functional analysis on Banach spaces to some metric spaces. Specifically, I wondered whether if one were to assume that the metric space possesses some parametrized segment-like structure (which, for example, in Banach spaces would consist of functions of the form $[0,1] \ni t \mapsto (1-t)x + ty$, where $x, y$ are elements of some Banach space), we could define something analogous to continuous linear (or, more appropriately, affine) operators by requiring that such operator would "preserve" this structure.

I'd like to ask if there are any papers or books which might be related to this topic and, if so, where should I look for them.

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  • $\begingroup$ You may consult one of the chapters of "W. H., Lattices with real numbers as additive operators, Dissertationes Mathematicae LXII, Warszawa 1969 -- it shouldn't hurt. $\endgroup$
    – Wlod AA
    Commented Jul 27, 2021 at 5:10
  • $\begingroup$ Algebra associates with operations. In the case of a metric space, I would consider relations, not operations. This could result in a semi-algebraic style. $\endgroup$
    – Wlod AA
    Commented Jul 27, 2021 at 5:17
  • $\begingroup$ This post seems to be related mathoverflow.net/questions/92755/… $\endgroup$
    – Onur Oktay
    Commented Jul 27, 2021 at 6:03
  • $\begingroup$ @KacperKurowski There is an insightful summary of the Closed Graph theorem(s) by Terrence Tao that you may find helpful. terrytao.wordpress.com/2012/11/20/… $\endgroup$
    – Onur Oktay
    Commented Jul 27, 2021 at 17:52

3 Answers 3

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An early attempt was made in:

Busemann, Herbert, Spaces with non-positive curvature. Acta Math. 80 (1948), 259–310.

Take a look at this paper by Gelander-Karlsson-Margulis for a modern treatment.

Hadamard observed that non-positively curved Riemannian manifolds share certain properties with Hilbert spaces. This theme was later extended to general metric spaces which satisfy the CAT(0) axiom. In a similar fashion, some properties of Banach spaces are shared by non-positively curved Finsler manifolds and Busemann's axiom extends this to general metric spaces. Roughly, the axiom is that the space is geodesic and the distance function between two geodesics is convex. Of course, more details are given in the references I shared.

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Embedding metric spaces into (larger) Banach spaces may suit to your purpose. Every metric space $X$ can be isometrically embedded into the Banach space $C(X)$ of continuous functions on $X$.

Let $x_0\in X$ be a fixed point. For each $x\in X$, define $f_x\in C(X)$ by $$f_x(u) = d(x,u)-d(x_0,u).$$ It is not hard to show that $\|f_x\|=d(x,x_0)$ and $\|f_x-f_y\|=d(x,y)$. The map $f:X\to C(X)$, $f(x) = f_x$ is the isometric embedding.

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    $\begingroup$ While I see that the entirety of a metric structure would be encoded in such an embedding, I'm not sure whether it would preserve the mentioned "segmental" structure in a sense that this embedding might not map "segments" in a metric space to segments in a normed space. Because of that, I worry that such an embedding might obfuscate some structure that might be more apparent if we were to look at the metric space in a more intrinsic way. $\endgroup$
    – Kacper Kurowski
    Commented Jul 26, 2021 at 23:21
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    $\begingroup$ Generally metric spaces do not possess such segments as you mention, unless you assume some extra structure. Thus, the embedding above provides you a way to define linear segments. On the other side, if there is a manifold structure (take torus as an example) on your metric space, you may choose to work with geodesics. Geodesics are analogous to linear segments in the way that they minimize the distance between given two points. It might be interesting to study the functions that maps geodesics onto geodesics. $\endgroup$
    – Onur Oktay
    Commented Jul 27, 2021 at 0:29
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    $\begingroup$ @OnurOktay, this is a very well-known embedding but I don't think that this one does anything algebraic to the given metric space. $\endgroup$
    – Wlod AA
    Commented Jul 27, 2021 at 5:15
  • $\begingroup$ @WlodAA except it embeds the metric space $X$ in a larger metric space with an algebraic (vector space) structure. If $X$ already possess some other algebraic structure, for example if $X$ is Lie group, then the embedding above would not preserve this extra structure. In that case, perhaps 1-parameter subgroups & continuous group homomorphisms are what one might want to study. Various theorems of functional analysis has analogous versions for continuous homomorphisms on Lie groups. $\endgroup$
    – Onur Oktay
    Commented Jul 27, 2021 at 5:55
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(I could post this answer also under Is there an algebraic approach to metric spaces?)

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While it is difficult, virtually impossible, to define metrically meaningful operations $\ f:X^n\to X\ $ over all metric spaces, it is possible to define something as (non-linear) functionals, say $\ f:X^n\to \mathbb R\ $ -- they may give the theory of the metric space some algebraic flavor.

Thus, let's consider an arbitrary metric space $\ (X\,\ d).\ $ Define, for all $\ x\ y\ \ldots \in X,$

  1. $\ \tau(x\ y\ z)\ :=\ d(x\ y)+d(y\ z)-d(x\ z)\ $ -- the triangle inequality;
  2. $\ \rho(x\ y\ z)\ $ and $\ \kappa(x\ y\ z)\ $ -- the radius and curvature of the Euclidean triangle isometric to the triangle $\ \{x\ y\ z\}\subseteq X;$
  3. $\ \pi(x\ y\ z)\ :=\ d^2(x\ y)+d^2(y\ z)-d^2(x\ z)\ $ -- sharp/right/obtuse angle
  4. e.t.c.

Already, the above three metric functionals allow extending a large part of Euclidean geometry over arbitrary spaces.

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