Metric Spaces and ${\mathbb{R}}^n$

The bridge from one-variable to multivariable analysis (analysis spine). The notions of limit, continuity, and completeness developed for $\mathbb{R}$ all depend only on the distance $|x-y|$. Abstracting distance to a metric lets us run the same arguments in ${\mathbb{R}}^n$, in function spaces, and on manifolds. The page culminates in the contraction mapping theorem, the fixed-point engine behind the inverse and implicit function theorems (inverse-implicit.md).

This page is the analysis-side companion to the point-set topology in topology-manifolds.md §1; that page works with general open-set topologies, this one with the metric that generates them.

References: Rudin, Principles of Mathematical Analysis, ch. 2; Munkres, Topology.

1. Metric Spaces

A metric space is a set $X$ with a function $d:X\times X\to [0,\infty )$ (the metric) such that for all $x,y,z$:

• (M1) $d(x,y)=0⟺x=y$;
• (M2) $d(x,y)=d(y,x)$ (symmetry);
• (M3) $d(x,z)\le d(x,y)+d(y,z)$ (triangle inequality).

Examples: $\mathbb{R}$ with $d(x,y)=|x-y|$; ${\mathbb{R}}^n$ with the Euclidean metric; the space $C[a,b]$ of continuous functions with the sup metric $d(f,g)={\sup }_x|f(x)-g(x)|$ — in which "convergence" means uniform convergence (sequences-series.md §7).

2. Norms on ${\mathbb{R}}^n$

${\mathbb{R}}^n$ is a vector space; a norm $\Vert \cdot \Vert$ assigns lengths with $\Vert x\Vert =0⟺x=0$, $\Vert \alpha x\Vert =|\alpha |\Vert x\Vert$, and $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert$, and induces the metric $d(x,y)=\Vert x-y\Vert$. The standard norms:

$$\Vert x{\Vert }_2=(\sum _i{x}_i^2{)}^{1/2},\Vert x{\Vert }_1=\sum _i|x_i|,\Vert x{\Vert }_{\infty }=\underset{i}{\max }|x_i|.$$

The Euclidean norm comes from the inner product $⟨x,y⟩={\sum }_i{x}_i{y}_i$, with the Cauchy–Schwarz inequality $|⟨x,y⟩|\le \Vert x{\Vert }_2\Vert y{\Vert }_2$.

All norms on ${\mathbb{R}}^n$ are equivalent: for any two norms there are constants $0<c\le C$ with $c\Vert x{\Vert }_a\le \Vert x{\Vert }_b\le C\Vert x{\Vert }_a$. Consequently open sets, convergence, continuity, and compactness in ${\mathbb{R}}^n$ do not depend on which norm is chosen. (This fails in infinite dimensions.)

3. Open Sets, Limits, Continuity

The open ball is $B_r(x)=\{y:d(x,y)<r\}$. A set $U$ is open if every point has a ball inside $U$; closed if its complement is open. These open sets form a topology — the metric topology — and recover exactly the structure of topology-manifolds.md §1.

• A sequence $x_n\to x$ iff $d(x_n,x)\to 0$. In ${\mathbb{R}}^n$ this is equivalent to coordinatewise convergence.
• $f:X\to Y$ between metric spaces is continuous at $x$ iff $\forall \epsilon \exists \delta :d_X(x,x^{\prime })<\delta \Rightarrow d_Y(f(x),f(x^{\prime }))<\epsilon$ — equivalently, $f^{-1}(\text{open})$ is open (continuity.md §2.1). Both the $\epsilon$–$\delta$ and the topological definitions transfer verbatim.

4. Completeness

A sequence is Cauchy if $d(x_m,x_n)\to 0$ as $m,n\to \infty$. A metric space is complete if every Cauchy sequence converges.

• $\mathbb{R}$ is complete (Cauchy criterion, sequences-series.md §4).
• ${\mathbb{R}}^n$ is complete: a Cauchy sequence is Cauchy in each coordinate.
• $C[a,b]$ with the sup metric is complete — this is just the statement that a uniform limit of continuous functions is continuous. Complete normed spaces are Banach spaces.

Completeness is the property that guarantees "the limit you are trying to build actually exists," and it is the hypothesis the contraction mapping theorem needs.

5. Compactness

$K\subseteq X$ is compact if every open cover has a finite subcover (the definition from topology-manifolds.md §1.4). In metric spaces this is equivalent to sequential compactness (every sequence has a convergent subsequence) and to being complete and totally bounded.

Heine–Borel Theorem. A subset of ${\mathbb{R}}^n$ is compact iff it is closed and bounded.

Proof idea. Boundedness lets Bolzano–Weierstrass (sequences-series.md §3) extract a convergent subsequence coordinatewise; closedness keeps the limit inside. $\square$

Consequences used throughout the spine: a continuous real function on a compact set attains its extrema (EVT, continuity.md) and is uniformly continuous (Heine–Cantor) — the latter is what makes continuous functions integrable over boxes in multiple-integrals.md. Warning: Heine–Borel is special to finite dimensions; closed bounded sets in $C[a,b]$ need not be compact.

6. Connectedness

$X$ is connected if it is not a disjoint union of two nonempty open sets, and path-connected if any two points are joined by a continuous path. For open subsets of ${\mathbb{R}}^n$ the two coincide. Connectedness is what makes the "constant if derivative vanishes" and "path-independence of conservative fields" arguments work (vector-calculus.md §3). An open connected set is called a domain or region.

7. The Contraction Mapping Theorem

A map $T:X\to X$ on a metric space is a contraction if there is a constant $0\le k<1$ with

$$d(Tx,Ty)\le kd(x,y)\text{for all }x,y.$$

Banach Fixed Point Theorem. A contraction on a nonempty complete metric space has a unique fixed point $x^{\ast }=T{x}^{\ast }$, and for any $x_0$ the iterates $x_{n+1}=T{x}_n$ converge to $x^{\ast }$ with $d(x_n,x^{\ast })\le \frac{k^n}{1-k}d(x_1,x_0)$.

Proof. The iterates are Cauchy: $d(x_{n+1},x_n)\le k^{n}d(x_1,x_0)$, and a geometric-series estimate bounds $d(x_m,x_n)$. By completeness $x_n\to x^{\ast }$; continuity of $T$ gives $T{x}^{\ast }=x^{\ast }$. Uniqueness: if $T{x}^{\ast }=x^{\ast }$ and $T{y}^{\ast }=y^{\ast }$ then $d(x^{\ast },y^{\ast })\le kd(x^{\ast },y^{\ast })$ forces $d=0$. $\square$

This theorem is the workhorse behind the inverse and implicit function theorems (inverse-implicit.md), the Picard–Lindelöf existence theorem for ODEs, and many numerical iteration schemes. It is precisely why completeness was worth axiomatizing.

8. Where this page is used

• Norms and Cauchy–Schwarz set up the linear algebra of derivatives in multivariable.md.
• Compactness (Heine–Borel) underlies integrability over boxes in multiple-integrals.md.
• Contraction mapping proves the inverse/implicit function theorems in inverse-implicit.md.
• Open-set / continuity machinery is the metric instance of the general topology in topology-manifolds.md.

Next: Multivariable Differentiation.