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The 2-adic absolute value is the p-adic absolute value for the prime number $p=2$. It is defined on the rational numbers by $|x{|}_2=2^{-v_2(x)}$ for $x\ne 0$ and $|0{|}_2=0$, and extends to their P-adic number completion ${\mathbb{Q}}_2$. For a nonzero integer, count factors of two until the remaining factor is odd; for a rational number, subtract the denominator's count from the numerator's count.

The associated metric reverses a familiar comparison:Although $13$ is eight units from $5$ on the ordinary number line, it agrees with $5$ modulo $8$, whereas $6$ already disagrees modulo $2$. The 2-adic absolute value measures that agreement. The value $|3/8{|}_2=8$ also shows why this Non-Archimedean absolute value must be distinguished from the ordinary absolute value $|3/8|=3/8$.

In the left panel, differences farther from the diagonal grow in ordinary absolute value. In the right panel, nonzero even differences are closer than odd differences, and each further factor of two halves the 2-adic absolute value. The highlighted pairs have opposite distance rankings. The repeated pattern is the nested congruence class structure of an ultrametric space, visible without adding any new points.

The loss $|a-\theta |$ for estimating a real parameter $\theta$ by $a$. Its risk of a decision rule is the corresponding expected value. Unlike squared-error loss, its pointwise size is a metric, so the triangle inequality gives a direct overlap version of the Le Cam two-point lemma.

This metric identifies $\mathbb{R}$ isometrically with the open interval $(-\pi /2,\pi /2)$ in the usual real distance. It induces the usual topology but is incomplete: the integer sequence has transformed limit $\pi /2$ outside that interval. This illustrates that completeness depends on the metric, not solely on its topology.

A Banach space is a normed vector space over the real numbers or complex numbers that is complete for the metric $d(x,y)=\Vert x-y\Vert$. Explicitly, whenever a sequence $(x_n)$ satisfiesthere is an element $x$ of the same space with $\Vert x_n-x\Vert \to 0$. The hypothesis says that the approximations eventually agree to any requested accuracy; completeness says that they describe an actual object in the space. The Cauchy sequence condition can be checked before anyone knows what that object is.

This makes a Banach space a natural setting for constructing solutions by successive improvements. If corrections $v_k$ have ${\sum }_k\Vert v_k\Vert <\infty$, their partial sums form a Cauchy sequence by the triangle inequality, and completeness supplies their sum. In fact, convergence of every such absolutely summable vector series characterizes a Banach space. The contraction mapping theorem uses precisely this mechanism: each correction is smaller than the preceding one by a fixed factor.^[ref]

Every normed vector space of finite dimension is a Banach space. Infinite-dimensional examples include the absolutely summable sequence space and the l-infinity sequence space; the choice of norm remains part of the specification. A collection of increasingly accurate finite descriptions can require an infinite description for its limit, as the finitely supported sequence example below demonstrates. Completion of a normed space adds these missing limits while preserving all original distances.

The Bhattacharyya distance is the negative natural logarithm of the Bhattacharyya coefficient, with $-\ln 0=\infty$. It is zero for identical probability distributions and infinite for disjoint probability supports. Despite the name, it need not satisfy the triangle inequality required of a metric.

A group distance is bi-invariant when both left and right translations are isometries. Conjugation then fixes the identity element and is an isometry, so each metric ball there is invariant under every inner automorphism. A compatible bi-invariant group metric therefore requires arbitrarily small conjugation-invariant neighbourhoods. This concerns a distance function, not merely a bi-invariant Riemannian metric tensor.

A Hausdorff topological group admits a compatible left-invariant group metric exactly when it has a countable neighbourhood basis at the identity element. Choose symmetric neighbourhoods of the identity element $U_n$ with $U_{n+1}^3\subset U_n$ and trivial intersection. Assign a step in $U_n$ cost $2^{-n}$, and take the infimum of total costs of chains joining two points. The dyadic product estimate for group neighbourhoods makes this a genuine compatible metric. The Hausdorff assumption separates distinct points; without it the same construction can give only a pseudometric.

The bounded-Lipschitz metric tests probability measures against a uniformly bounded Lipschitz class. It metrizes weak convergence of probability measures on separable metric spaces. With a complete compatible base metric, it is complete on the probability measures of a Polish space. Replacing the sum norm by the maximum norm changes the metric only by uniform constant factors.

The increasing function $\varphi (s)=s/(1+s)$ is subadditive on nonnegative arguments: each summand $b/(1+b)$ and $c/(1+c)$ is at least its numerator divided by $1+b+c$. Thus $\varphi \circ d$ is a metric whenever $d$ is. It is bounded and induces the same topology because $\varphi$ and its inverse near zero preserve convergence to zero.

The rational numbers $q_n=\lfloor 10^n\sqrt{2}\rfloor /10^n$ lie in $[1,2]$ and obey $0\le \sqrt{2}-q_n<10^{-n}$. They form a convergent sequence in the real numbers, with limit of a sequence $\sqrt{2}$. Every subsequence has that same real limit. Because $\sqrt{2}$ is an irrational number, no subsequence converges to a point in the rational numbers with their usual metric. Boundedness alone therefore does not give the corresponding theorem in every metric space; completeness of the real numbers is doing real work here. The floor function simply truncates the decimal expansion in this example.

For a continuous map on a compact metric space, the Bowen metric compares the first $n$ points of two forward orbits. Its radius-$ϵ$ ball is $B_n(x,ϵ)=\{y:d_n(x,y)<ϵ\}$. If $s_n(ϵ)$ is the largest cardinality of an $ϵ$-separated set in this metric, then topological entropy is ${\lim }_{ϵ\downarrow 0}{\mathrm{limsup}}_{n\to \infty }{n}^{-1}\log s_n(ϵ)$. The uniform continuity of the identity between two compatible metrics gives the comparison of separated-set counts needed for independence of the original metric.

Cantor space is the binary sequence space with the product topology of discrete two-point factors. Here the Bernoulli-space synonym refers to this topology, without selecting a particular probability measure. The metric $d(x,y)={\sum }_{j\ge 1}{2}^{-j}|x_j-y_j|$ induces the topology. A diagonal subsequence argument gives compactness. Fixing a finite prefix gives a clopen cylinder set. The map $x\mapsto {\sum }_{j}2x_j{3}^{-j}$ is a homeomorphism to the usual Cantor set.

A Cauchy sequence is a sequence whose sufficiently late terms are all close to one another. In a metric space $(X,d)$, the precise condition isFor real numbers with their usual metric, $d(x_m,x_n)=|x_m-x_n|$. The useful feature is that the definition compares computed terms to other computed terms: it does not require an already known limit of a sequence. The one index $N$ must control every pair of later indices, not merely successive terms. ^[ref].

For an explicit example, take $q_n=1-2^{-n}$ for $n\ge 1$. Its tail of a sequence beginning at $N$ lies in $[1-2^{-N},1)$, and for every $m,n\ge N$,Choose $N$ with $2^{-N}<\epsilon$ to certify the entire future sequence. For $\epsilon =0.1$, $N=4$ works because $2^{-4}=0.0625$. The limit of a sequence is one, but the pairwise bound establishes the Cauchy sequence property without first needing that answer. By contrast, $a_n=(-1{)}^n$ is not a Cauchy sequence: arbitrarily late even integers and odd integers give values separated by two. ^[ref].

Tiny successive changes do not provide this guarantee. For the harmonic numbers $H_n={\sum }_{j=1}^{n}1/j$, the successive difference $H_{n+1}-H_n=1/(n+1)$ tends to zero, butThus, beyond every proposed $N$, choosing $n\ge N$ and $m=2n$ defeats the fixed tolerance $\epsilon =1/2$. Even every fixed positive lag $h$ passes the weaker test: $0<H_{n+h}-H_n\le h/(n+1)\to 0$. The failing lag grows with $n$, as explained more generally by convergence from arbitrary variable-lag increments. ^[ref].

Every convergent sequence in a metric space is a Cauchy sequence: apply the triangle inequality to two late terms and their common limit of a sequence. The reverse implication requires completeness. It holds in the real numbers by completeness of the real numbers, but can fail in the rational numbers: the rational endpoints from the bisection method approaching $\sqrt{2}$ are a Cauchy sequence with no rational limit of a sequence. The existing construction in completeness of the real numbers shows how this happens. A Cauchy sequence is always a bounded sequence; in $\mathbb{R}$, boundedness gives a convergent subsequence by the Bolzano-Weierstrass theorem, and the pairwise condition forces the whole sequence to share its limit. ^[ref].

The metric induced by the maximum norm measures the largest difference in Cartesian coordinates. Simultaneous changes of Cartesian coordinates cost only as much as the slowest coordinate requires.

A weakly closed uniformly tight family of Borel probability measures on a Polish space is weakly compact. Embed the underlying space into a compact metric closure, take subsequential pushforward limits, and use the closed-set Portmanteau theorem on the common tightness compact sets. The limit retains full mass in the original space; inverse continuity of the probability pushforward embedding theorem then returns the weak limit.