Complete metric space

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In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M or alternatively if every Cauchy sequence in M converges in M.

Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). Thus, a complete metric space is analogous to a closed set. For instance, the set of rational numbers is not complete, because e.g. \sqrt{2} is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it. (See the examples below.) It is always possible to "fill all the holes", leading to the completion of a given space, as explained below.

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Examples

The space Q of rational numbers, with the standard metric given by the absolute value, is not complete. Consider for instance the sequence defined by x1 := 1 and xn+1 := xn/2 + 1/xn. This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: Such a limit x of the sequence would have the property that x2 = 2, but no rational numbers have that property. But considered as a sequence of real numbers R it converges towards the irrational number \sqrt{2}, the square root of two.

The open interval (0,1), again with the absolute value metric, is not complete either. The sequence (1/2, 1/3, 1/4, 1/5, ...) is Cauchy, but does not have a limit in the space. However the closed interval [0,1] is complete; the sequence above has the limit 0 in this interval.

The space R of real numbers and the space C of complex numbers (with the metric given by the absolute value) are complete, and so is Euclidean space Rn. In contrast, infinite-dimensional normed vector spaces may or may not be complete; those that are, comprise the Banach spaces. The space C[a,b] of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C(a,b) of continuous functions on (a,b), for it may contain unbounded functions. Instead, with the topology of compact convergence, C(a,b) can be given the structure of a Fréchet space: a locally convex topological vector spaces whose topology can be induced by a complete translation-invariant metric.

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