Complete Metric Space

The expansion constant of x d is 2.

Complete metric space. Any convergent sequence in a metric space is a cauchy sequence. A metric space x d is complete if any of the following equivalent conditions are satisfied. The hope with this initial construction is that c e d is a complete metric space but as will be seen in part v of exercise 1 2 dfails to even be a metric.

The resulting space will be denoted by xand will be called the completion of xwith respect to d. In the same sense one understands the completeness of a pseudo metric space and a space with a symmetry. They are called uniformic uniformly isomorphic if there exists a uniform isomorphism between them i e a bijection.

A sequence x n in x is called a cauchy sequence if for any ε 0 there is an n ε n such that d x m x n ε for any m n ε n n ε. Assume that x n is a sequence which converges to x. They are called homeomorphic topologically isomorphic if there exists a homeomorphism between them i e a bijection.

Let x d be a metric space. Let ε 0 be. A complete metric space is a particular case of a complete uniform space.

The goal of these notes is to construct a complete metric space which contains x as a subspace and which is the smallest space with respect to these two properties. Complete metric spaces definition 1. Examples include the real numbers with the usual metric the complex numbers finite dimensional real and complex vector spaces the space of square integrable functions on the unit interval l 2 0 1 and the p adic numbers.

A metric space is called complete if each cauchy sequence in it converges. A subset of a complete metric space is itself a complete metric space if and only if it is closed. A closed subset a of a complete metric x d space is itself a complete metric space with the distance which is the restriction of d to a.