Banach-like analysis on metric spaces 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.
 A: 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.
A: 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.
A: (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 $\ x\ y\ \ldots \in X,$

*

*$\ \tau(x\ y\ z)\ :=\ d(x\ y)+d(y\ z)-d(x\ z)\ $ -- the triangle inequality;

*$\ \rho(x\ y\ z)\ $ and $\ \kappa(x\ y\ z)\ $ -- the radius and curvature of the Euclidean triangle isometric to the triangle $\ \{x\ y\ z\}\subseteq X;$

*$\ \pi(x\ y\ z)\ :=\ d^2(x\ y)+d^2(y\ z)-d^2(x\ z)\ $ -- sharp/right/obtuse angle

*e.t.c.

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