Metric Spaces, Norms, and Examples

Every limit rests on a single number: the distance between two things. A sequence of reals converges to $L$ when $|x_n-L|$ eventually stays small; a function is continuous at $c$ when $f(x)$ is close to $f(c)$ whenever $x$ is close to $c$. In each case close means only that $|x-y|$ is small. If some other set carries a sensible notion of distance, every argument about limits transfers to it without change.

A metric space is a set together with such a distance function, stripped to the properties the proofs use. One definition and one set of theorems then govern sequences in ${\mathbb{R}}^n$, convergence of functions, points on a sphere, and spaces whose elements are themselves functions, so nothing about limits has to be re-proved in each new setting.¹

The metric axioms

The first three axioms encode obvious demands on a notion of distance: it is never negative, only a point and itself are at distance zero, and the distance from $x$ to $y$ equals the distance back. The triangle inequality (M4) is the one with content. It says a detour through $y$ can never be a shortcut from $x$ to $z$, and essentially every estimate in analysis is an application of it.

A useful consequence, obtained by applying (M4) twice, is the reverse triangle inequality:

It bounds how much a distance can change when one endpoint moves, and is the metric-space form of the estimate $||a|-|b||\le |a-b|$ from the real line.

Examples

The verifications are mechanical for (M1)–(M3); the triangle inequality is the only step that ever takes work.

The real line. On $X=\mathbb{R}$, set $d(x,y)=|x-y|$. Axioms (M1)–(M3) are immediate, and (M4) is the ordinary triangle inequality for absolute value:

This is the standard metric on $\mathbb{R}$; unqualified, $\mathbb{R}$ means this space.

Euclidean space. On $X={\mathbb{R}}^n$, with points $x=(x_1,\dots ,x_n)$, the standard metric is

Only (M4) is delicate. It follows from the Cauchy–Schwarz inequality, worth isolating because it recurs throughout analysis.²

The lemma comes from expanding the nonnegative double sum ${\sum }_{k,\ell }(x_k{y}_{\ell }-x_{\ell }{y}_k{)}^2\ge 0$ and relabelling.

The taxicab and sup metrics. The same set ${\mathbb{R}}^n$ carries other metrics. The taxicab (or ${\ell }^1$) metric sums coordinate distances; the sup (or ${\ell }^{\infty }$) metric takes the largest:

Both satisfy all four axioms; their triangle inequalities reduce term-by-term to the scalar case. On ${\mathbb{R}}^n$ these three are the working examples, and they are genuinely different distance functions on the same underlying set.

The discrete metric serves as a stress test: a claim about metric spaces that fails under it is false in general. It also shows a set with no numerical structure can still be a metric space.

A bounded remetrization of $\mathbb{R}$. The function

is a metric on $\mathbb{R}$ under which every pair of points is less than $1$ apart. It is $\phi (|x-y|)$ for the increasing, subadditive $\phi (t)=t/(1+t)$, and subadditivity of $\phi$ gives (M4). The same set can thus be bounded under one metric and unbounded under another, so boundedness is a property of the metric, not of the set alone.

Continuous functions. Let $C[a,b]$ be the set of continuous $f:[a,b]\to \mathbb{R}$, with the sup metric $d_{\infty }(f,g)={\sup }_{t\in [a,b]}|f(t)-g(t)|$.

Two functions are close in this metric exactly when their graphs stay uniformly near each other, so the largest vertical gap between the graphs is the distance.

Treating a whole set of functions as points in one metric space is what makes uniform convergence, and fixed-point arguments such as Picard's theorem, a matter of convergence in a single space.

The sphere. On the unit sphere $S^2=\{x\in {\mathbb{R}}^3:x_1^2+x_2^2+x_3^2=1\}$, the great-circle distance takes $d(x,y)$ to be the angle $\theta \in [0,\pi ]$ between the radii to $x$ and $y$, computable as $d(x,y)=\arccos (x_1{y}_1+x_2{y}_2+x_3{y}_3)$. The first three axioms are clear; the triangle inequality is the spherical law of cosines. This is the distance used on the surface of the earth, and it is not the restriction of the Euclidean metric of ${\mathbb{R}}^3$ (that would be the straight-line chord through the interior). The same point set thus supports two natural, different metrics.

The discrete metric as a source of counterexamples

The discrete metric is extreme, and any claim that looks obvious from Euclidean pictures should be checked against it. A ball is either a single point or the whole space; every set is bounded and closed; and there is no notion of between. When a statement about metric spaces holds for Euclidean space but fails under the discrete metric, the Euclidean intuition was using structure the axioms do not provide.

Norms

Most of these metrics measure distance as the size of a difference vector. Abstracting that pattern gives a norm, the source of nearly every metric on a vector space.³

Every norm induces a metric by measuring the size of the difference.

Nonnegativity and (M2) come from the first norm axiom; symmetry from homogeneity with $\alpha =-1$; and (M4) from the norm's triangle inequality, since $x-z=(x-y)+(y-z)$. The three metrics on ${\mathbb{R}}^n$ arise this way from the ${\ell }^p$ norms

with $p=1$ giving taxicab, $p=2$ Euclidean, and the limiting case the sup norm. The metrics differ, but their unit balls show the difference most clearly: the set $\{x:\Vert x\Vert \le 1\}$ is a diamond for ${\ell }^1$, the round disk for ${\ell }^2$, and a square for ${\ell }^{\infty }$.

The nesting $\{\Vert x{\Vert }_{\infty }\le 1\}\supseteq \{\Vert x{\Vert }_2\le 1\}\supseteq \{\Vert x{\Vert }_1\le 1\}$ reflects the pointwise inequality $\Vert x{\Vert }_{\infty }\le \Vert x{\Vert }_2\le \Vert x{\Vert }_1$ on ${\mathbb{R}}^n$. On a finite-dimensional space these norms also satisfy reverse bounds like $\Vert x{\Vert }_1\le \sqrt{n}\Vert x{\Vert }_2$, so any two are within constant factors of each other.

The three ${\ell }^p$ metrics on ${\mathbb{R}}^n$ are pairwise strongly equivalent. Strong equivalence forces a ball of one metric to sit inside a ball of the other and vice versa, so the two metrics have the same open sets, the same convergent sequences, and the same continuous functions, even though they assign different numerical distances. Properties unchanged by remetrization (open, convergent, compact) are topological; properties that do change (the numerical value of a distance, whether a specific radius works) are metric-specific. The distinction recurs when the topology is built.

Balls, boundedness, and diameter

Every later definition is built from one geometric object, the ball: all points within a fixed distance of a center.

The shape of a ball depends entirely on the metric. In $\mathbb{R}$ with the standard metric, $B(x,\delta )=(x-\delta ,x+\delta )$ is an open interval. On ${\mathbb{R}}^2$ the unit balls of the three norms above are the diamond, disk, and square. Under the discrete metric on any $X$, $B(x,\delta )=\{x\}$ for $\delta \le 1$ and $B(x,\delta )=X$ for $\delta >1$: the balls have only two possible shapes.

Boundedness is expressed through balls: a set is bounded when it fits inside one.

For subsets of $\mathbb{R}$ this recovers the earlier notion of a bounded set: every element lies within a fixed distance of a reference point. The choice of reference point $p$ does not matter, since moving it changes the required $B$ by only $d(p,p^{\prime })$ by the triangle inequality. Under the discrete metric every set has diameter at most $1$, so every set is bounded, whatever its size.

Subspaces

Any subset of a metric space is a metric space in its own right, by simply restricting $d$.

Balls in a subspace are computed with the restricted metric, so they can differ from the ambient balls. In $[0,1]$ as a subspace of $\mathbb{R}$,

which includes its left endpoint, unlike $B_{\mathbb{R}}(0,\frac{1}{2})=(-\frac{1}{2},\frac{1}{2})$. The ball is always computed in whichever space is the ambient one, and the answer can change when the ambient space does.

Properties like open and closed are likewise relative to the surrounding space, made precise once the topology is built from these balls.