Complete Metric Space

Cauchy Sequence

Cauchy Sequence

Let be a metric space. A sequence in is called a Cauchy sequence if for any there exists such that for all .

Theorem

Convergent sequences in metric spaces are Cauchy sequences.

Proof Suppose is convergent with . Then for any , there exists such that for all . Therefore, for all , we have Thus is a Cauchy sequence.

Proposition

Cauchy sequences in metric spaces are bounded.

Proof Let be a Cauchy sequence. Then there exists such that for all . Let Then for all n. Therefore for all , i.e. is bounded.

Proposition

Let be a metric space and let be a Cauchy sequence in . If has a convergent subsequence, then is convergent.

Proof Let be a Cauchy sequence and let be a convergent subsequence of with . We have for sufficiently large . By taking and using , we obtain Since is Cauchy, the right hand side goes to as . Therefore as , that is .

Theorem

Every Cauchy sequence in is convergent.

Proof Let be a Cauchy sequence in . Then is bounded. Thus, it follows from Bolzano-Weierstrass theorem that has a convergent subsequence. Consequently, by invoking previous theorem, is convergent.

Completeness

Complete Metric Space

A metric space is called complete if every Cauchy sequence in is convergent.

e.g.

• with the standard Euclidean metric is complete.
• with the induced Euclidean metric is not complete.
• is complete under the metric
  

Proposition

Let be a complete metric space and . Then is complete iff is closed in .

Proof Let be a sequence in with . Then is a Cauchy sequence in and hence is a Cauchy sequence in . Since is complete, for some . By uniqueness of limit, we have and thus . This shows that is closed by corollary. Conversely, let be a Cauchy sequence in . Then is also a Cauchy sequence in . Since is complete, for some . Since is closed, . Therefore is complete.

Universal Property of Completion

Every metric space has a unique completion, i.e. there exists a complete metric space and an injection such that and .

Contraction Mapping Theorem

Fixed Point

Let be a metric space and a map. A point is called a fixed point of if .

Contraction

Let be a metric space and a map. is called a contraction if there exists such that where is called the contraction rate of .

Contraction Mapping Theorem

Let be a complete metric space. If is a contraction, then it has a unique fixed point.

Quotient of Complete Metric Spaces

Proposition

Suppose is a complete metric space and is a compact subspace. Then the topological quotient is metrizable, and complete under the metric

Proof We first show that is a well-defined metric on . For and , we have and . Therefore, , which shows that the metric is well-defined. The other properties of a metric (non-negativity, symmetry, and triangle inequality) can be verified similarly.
Now we show that the topology induced by this metric coincides with the quotient topology by showing that for all .