Metric space and completeness: Definition & Examples

A metric space is a set together with a rule for measuring distance between its elements. It is complete when every Cauchy sequence—one whose sufficiently late terms become arbitrarily close to one another—has a limit inside that set. In quant work, this distinction matters when proving that repeated pricing, calibration or value-function updates converge to an admissible object: an iteration can become arbitrarily stable while approaching a point the chosen domain excludes.

Choose the distance as well as the domain

Write a metric space as $(X,d)$ . Its distance function $d:X\times X\to[0,\infty)$ must satisfy, for every $x,y,z\in X$ :

Requirement	Mathematical statement
Non-negative distance	$d(x,y)\geq0$
Zero distance identifies the same point	$d(x,y)=0\iff x=y$
Symmetry	$d(x,y)=d(y,x)$
Triangle inequality	$d(x,z)\leq d(x,y)+d(y,z)$

These metric axioms allow much more than geometric points. The elements might be parameter vectors, functions or strings. A norm supplies a metric through

d(x,y)=\lVert x-y\rVert

when subtraction makes sense, but a general metric space needs no vector addition or scalar multiplication.

Consider two hypothetical two-asset portfolios with weights expressed as fractions of NAV:

\mathbf{w}=(0.6,0.4),\qquad \mathbf{v}=(0.4,0.6).

Both weight vectors sum to one. Distance using the Euclidean norm is

d_2(\mathbf{w},\mathbf{v}) =\sqrt{(0.6-0.4)^2+(0.4-0.6)^2} =\sqrt{0.08}\approx0.282843.

Using the Manhattan norm instead gives

d_1(\mathbf{w},\mathbf{v}) =\lvert0.6-0.4\rvert+\lvert0.4-0.6\rvert=0.4.

Both are valid metrics; they summarise a change in holdings differently. Neither is a monetary transaction cost or a return-risk estimate without an additional model. Coordinates with different units also require an explicit scaling choice.

A loss is not automatically a metric. Squared Euclidean distance fails the triangle inequality. On the real line, the squared distance from

0

2

4

, whereas travelling through

1

gives

1+1=2

. A useful squared-error objective can therefore fail to define the distance geometry assumed by a theorem.

Cauchy asks about the whole tail

Convergence specifies a destination: $x_n\to x\in X$ means $d(x_n,x)\to0$ . The Cauchy condition instead compares the sequence with itself:

\text{for every }\varepsilon>0,\quad \text{there exists }N\text{ such that } d(x_m,x_n)<\varepsilon \quad\text{for all }m,n\geq N.

Every convergent sequence is Cauchy. Completeness supplies the reverse implication: every Cauchy sequence converges to an element of $X$ , using the same metric. This is the distinction formalised in Eck's definitions and convergence theorem.

Checking only adjacent terms is weaker. For $x_n=\log n$ with ordinary absolute distance, $x_{n+1}-x_n=\log(1+1/n)\to0$ , but

x_{2n}-x_n=\log2

for every $n$ . Two terms arbitrarily far into the tail remain a fixed positive distance apart. Small successive updates alone do not prove a numerical routine has converged.

A missing boundary point breaks completeness

Equip $X=(0,1)$ with $d(x,y)=\lvert x-y\rvert$ . The sequence

x_n=2^{-n},\qquad n=1,2,\ldots

starts $0.5,0.25,0.125,0.0625,\ldots$ and stays inside $X$ . Choose $N$ so that $2^{-N}<\varepsilon$ . For all $m,n\geq N$ ,

\lvert2^{-m}-2^{-n}\rvert\leq2^{-N}<\varepsilon.

It is Cauchy. Its real-number limit is $0$ , which is outside $(0,1)$ , so this metric space is incomplete. No different point inside the interval can be its limit: for any fixed $p>0$ , $\lvert2^{-n}-p\rvert\to p$ , not zero.

The exact sequence 0.5, 0.25, 0.125 and smaller powers of one half approaches zero under absolute distance. The open interval (0,1) excludes zero; the closed interval from zero to one includes it.

The same Cauchy sequence and the same distance in two domains. Zero is missing from the first domain and included in the second. Completeness of the closed interval concerns every Cauchy sequence, not only the one drawn.

Enlarging the domain to $[0,1]$ admits this limit. More generally, a closed subset of a complete metric space is complete under the inherited distance. Since the real line is complete, this establishes completeness of $[0,1]$ for all its Cauchy sequences.

The example also explains a hypothesis of the Banach fixed point theorem. The map

T(x)=x/2

sends

(0,1)

into itself and satisfies

d(T(x),T(y))=\tfrac12 d(x,y).

Yet its only possible fixed point solves

x/2=x

, giving

x=0

, outside the domain. On

[0,1]

, the same map has that fixed point. The contraction theorem requires a nonempty complete metric space as well as a contraction mapping the space into itself. Contracting distances cannot supply a missing limit point.

Two distinctions to retain

Completeness depends on the metric. On the same set

(0,1)

, assign distance

1

to distinct points and

0

to identical points. This discrete metric makes every Cauchy sequence eventually constant: apply the Cauchy condition with

\varepsilon=1/2

. The space is then complete, even though it was incomplete under absolute distance.

Completeness is not compactness. In a metric space, compactness is equivalent to every sequence having a subsequence converging within the space. The real line with absolute distance is complete but not compact: the sequence

1,2,3,\ldots

has no convergent subsequence. Completeness governs Cauchy sequences; it does not make every sequence settle down.

For the underlying vector geometry, the Length and Triangle inequality sections of The Dot Product, Length, and Angle develop the Euclidean case. The public page is a preview; the full lesson requires membership.

Sources

David J. Eck, A Short Introduction to Metric Spaces, §5, “Completeness”: the Cauchy condition, convergence and completeness.
Jiří Lebl, Basic Analysis I, §7.1, “Metric spaces”: metric axioms, Euclidean and discrete metrics.
Jiří Lebl, Basic Analysis I, §7.4, “Completeness and compactness”: Cauchy completeness, closed subspaces and the distinction from compactness.
Jiří Lebl, Basic Analysis I, §7.6.1, “Fixed point theorem”: the contraction principle and the role of completeness.

Choose the distance as well as the domain

Cauchy asks about the whole tail

A missing boundary point breaks completeness

Two distinctions to retain

Sources

Related terms

Banach fixed-point theorem

Euclidean norm

Manhattan norm