Distances and Metric Spaces |

**In General**

* __Remark on terminology__: Following a common
usage in physics, I will reserve the word "metric" for a metric tensor (or tensor
field) *g*: *V* × *V* → \(\mathbb R\) on a vector space
*V*, while the type of function *d*: *X* × *X* →
\(\mathbb R\) on a set *X* defined below, that is often called "metric"
in mathematics, will be called a "distance" here.

* __Idea__: A distance is the most common way
of mathematically realizing the intuitive notion of closeness, although other definitions
are possible.

$ __Pseudometric space__: A pair (*X*,
*d*) with *X* a set and *d* a pseudodistance on *X*, a function
*d*: *X* × *X* → \(\mathbb R\) satisfying (1) *d*(*p*,
*q*) ≥ 0 and *d*(*p*, *p*) = 0, positive semi-definiteness;
(2) *d*(*p*, *q*) = *d*(*q*, *p*), symmetry;
(3) *d*(*p*, *q*) ≤ *d*(*p*, *r*)
+ *d*(*r*, *q*), triangle inequality;
> s.a. types of topologies.

* __Remark__: In such a "degenerate metric
space" two points *x* ≠ *y* with *d*(*x*, *y*)
= 0 have to be equidistant from all other *z*s, because of the triangle inequality,
and thus "indistinguishable"; This is not so in the Lorentzian case.

$ __Distance__: A positive-definite
pseudodistance *d*: *X* × *X* → \(\mathbb R\).

$ __Metric space__: A pair
(*X*, *d*), with *d* a distance on *X*.

* __Relationships__: It may arise from a
norm; In the topology induced by the distance, a metric space is always paracompact.

@ __References__: Kurepa 63;
Blumenthal 70;
Schreider 74;
Honig 95 [non-standard];
Deza & Deza 14 [encyclopedia].

@ __And physics__:
Goodson 16 [dynamical systems].

> __And other structures__:
see finsler geometry.

**Related Notions and Results**
> s.a. cover; entropy;
types of distances.

* __Interesting maps__: Isometries
and continuous maps (in the induced topology) are not so rich; Distance-decreasing
and *λ*-Lipschitz maps are more interesting.

* __Baire category theorem__:
A complete metric space is not the countable union of nowhere-dense sets;
This result can be stated as a theorem in Ramsey theory.

* __Dilation of a map__:
For *f* : *X* → *Y*, dil *f* :=
sup_{x ≠ x'} *d*(*f*(*x*),
*f*(*x*')) / *d*(*x*,*x*');
dil *f* := lim_{ε → 0}
dil *f* |_{B(x,ε)}.

* __Diameter__: If *A* ⊂ *X* is a
subset of a metric space, its diameter is diam(*A*):= l.u.b.{*d*(*x*, *y*)
| *x*, *y* in *A*}.

* __Equicontinuity__: A family
\(\cal F\) of functions on a metric space (*X*, *d*) is equicontinuous
iff for each *ε* > 0 there is a *δ* > 0
such that for all *x* and *x'* in *X*, and *f* in
\(\cal F\), *d*(*x*, *x'*) < *δ* implies
|*f*(*x*)−*f*(*x'*)| < *ε*.

$ __Outer measure of a set__:
The *d*-dimensional outer measure of *A* ⊂ *X* is
*m*_{d}(*A*):=
lim_{ε → 0}
inf ∑_{i} (diam
*S*_{i})^{d},
over all countable coverings of *A* by closed spheres
*S*_{i} of diameter < *ε*.

@ __Other structure on metric spaces__: Parthasarathy 67 [probability measures];
Penot JGP(07)
[tangent vectors and differentials of mappings].

**New Distances out of Old**

* __Sum and sup__: If
*d*_{i} are distances
on *X* (or even if all but one of them are pseudodistances),
then two new distances on *X* are

*d*(*x*, *y*):= ∑_{i}
*a*_{i}
*d*_{i}(*x*, *y*),
with *a*_{i} > 0 for all *i* ,
and* d*(*x*, *y*):=
sup_{i}
*d*_{i}(*x*, *y*) .

* __On subsets A of X__:
The induced distance

*

**Space of Metric Spaces** > s.a. distance between
manifolds with metrics; Gromov-Hausdorff Space.

@ __Measure__: Kondo DG&A(05)

**Generalizations**

* __Lorentzian metric space__:
A pair (*M*, *d*), with *d*: *M* × *M* →
\(\mathbb R\)_{+} ∪ {∞}, such that (i) *d*(*x*,
*y*) > 0 implies *d*(*y*, *x*) = 0, so in particular
*d*(*x*, *x*) = 0 for all *x*; (ii) *d*(*x*,
*y*) *d*(*y*, *z*) > 0 implies that *d*(*x*,
*z*) ≥ *d*(*x*, *y*) + *d*(*y*, *z*),
the "reverse triangle inequality"; Examples of Lorentzian distance are the
timelike geodesic distance between two points, or the volume of their Alexandrov set.

* __Probabilistic metric space__:
A generalization of a metric space, where the distance has values in a set of probability distribution functions;
> see Wikipedia page.

* __Quantum metric space__: A C*-algebra (or more generally an order-unit
space) equipped with a generalization of the Lipschitz seminorm on functions which is defined by an ordinary metric.

@ __Lorentzian distance__:
Erkekoglu et al GRG(03) [level sets];
in Noldus CQG(04)gq/03;
Rennie & Whale a1903 [finiteness and continuity];
> s.a. causality conditions; distance
on a manifold with metric; world function.

@ __Quantum metric space__:
Rieffel MAMS(04)m.OA/00
[Gromov-Hausdorff distance];
Latrémolière a1506 [Gromov-Hausdorff propinquity].

@ __Other__:
Schweizer & Sklar 83 [probabilistic metric spaces];
Mizokami & Suwada T&A(05) [and their resolutions];
Kopperman et al T&A(09) [partial metric spaces, completion];
Antoniuk & Waszkiewicz T&A(11) [duality of generalized metric spaces].

> __Bregman divergence__:
see Wikipedia page.

*A qué le llaman distancia, eso me habrán de explicar* − Atahualpa Yupanqui

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send feedback and suggestions to bombelli at olemiss.edu – modified 11 mar 2019