Complete metric space, DYK Fact #166

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A complete metric space is a metric space in which every Cauchy sequence converges.

Complete metric space: A metric space, in which every Cauchy sequence is convergent, is a complete metric space. Did you know? Examples include the real numbers with the usual metric, the complex numbers, finite-dimensional real and complex vector spaces, (and also the space of square-integrable functions on the unit interval L˛([0,1]), and the p-adic numbers). Let us notice however that Q is not a complete metric space, this leads thus to study an extension of Q to a completed space of Q.

Did you know?

The completeness of the real numbers is often formulated in terms of the least-upper-bound property: every set of real numbers that is bounded above has a real least upper bound. It turns out that this is equivalent to the condition that every real Cauchy sequence is convergent.

Let X be a metric space with metric d. Given a function f : X → X, suppose there exists a non-negative constant K < 1 such that for all x, y in X, d (f(x), f(y)) ≤ K d(x, y). (K can be 0, in which case f is constant.) The function f is called a contraction mapping on X.

Did you know? Banach’s fixed-point theorem states that if X is a complete metric space, any contraction mapping f on X has a unique fixed point, i.e. there exists exactly one point p ∊ X such that f(p) = p.

Did you know? Stefan Banach (1892–1945) was a Polish mathematician.

Did you know? Baire’s theorem for metric spaces states that if X is a complete metric space and is a countable collection of nowhere dense subsets of X, is everywhere dense in X. (A subset of a metric space X is nowhere dense iff the interior of its closure is empty; it is everywhere dense iff its closure is the whole space X.)

Did you also know? René Baire (1874–1932) was a French mathematician.

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Any metric space can be “completed”. In a metric space X with metric d, let us say that two Cauchy sequences and are equivalent iff . This is quite clearly an equivalence relation. Denote the collection of all the equivalence classes by [X]. If , ∊ [X], it can be shown that exists. Moreover, the mapping defined by can be shown to be well defined and is a metric on [X]. ([X], [d]), then, is a complete metric space containing a homeomorphic copy of (X, d).