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I'm reading a book on Metric Spaces and the author is always talking about the "geometry" of some metric spaces, but he doesn't say what he means by geometry.

For example:

Despite the fact that it is infinite-dimensional, the next example shares many nice geometric properties with the real line $\mathbb{R}$.

$\textbf{Example}$: $(l_\infty)$ This is the space whose elements consist of all bounded sequences $(x_1,x_2,..)$ of real numbers, with the distance $d_\infty (x,y)$ between two such sequences $x=(x_1,x_2,...)$ and $y=(x_1,x_2,...)$ taken as $d_\infty (x,y)=\sup\limits_{1\leq i <\infty}|x_i-y_i|$.

I think he is talking about some theorems that are true on both metric spaces and he considers to be of a "geometric flavor", but he gives no example of such theorems.

What does the author mean by "geometry" in this case? If I'm right, examples of such theorems would be nice.

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I think the example given by the author is a bad example to testify for his thesis. Here is a better one: The geometries on the euclidean plane and on the surface of a sphere are certainly different, but they share many interesting properties. For instance, given three (sufficiently small) lengths $a$, $b$, $c$ satisfying the triangle inequality there is, up to isometry, exactly one triangle having $a$, $b$, $c$ as side lengths.

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Geometry, i.e. the "measurement of the earth", in the case of metric spaces refers to the process of deducing information from measuring or characterizing the distance between objects in the given metric space.

Let us start with a caveat: a metric space is a rather primitive mathematical structure: by definition it is a set endowed with a function called distance (or metric). It is more primitive than a normed space or a space with scalar product as both these spaces need a linear space structure to be defined.

An example: in the familiar case of $\mathbb R^n$ the scalar product $\langle\cdot,\cdot\rangle$ induces the norm $\|\cdot\|$ which induces the distance $d(\cdot, \cdot)$. All 3 structures are widely used in every day computations. The resulting topology is metric and, in particular, $\mathbb R^n$ is a metric space. We can also talk about the sum of vectors and multipication by scalars though, as $\mathbb R^n$ is a vector space.

As remarked above, such richness of structures is peculiar to "nice" spaces. On the contrary, in many applications like clustering analysis for example, one works with finite sets and distances.

In summary, in the case of a metric space $(M, d)$ we are left to work with the distance function $d$ which, by definition, gives us the idea of "being close" or "far away". It can be used to define the concept of convergence of sequences, to define a topology (called metric topology), which is based on the concept of open ball $$B(x;r) = \{y \in M : d(x,y) < r\}.$$

A metric space can admit different distance functions; one can study the different geomtries induced by the distances. A good example is given by the taxicab geometry vs. the euclidean geometry in $\mathbb R^2$.

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