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In mathematical analysis, a metric space ''M'' is called complete (or a Cauchy space) 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). For instance, the set of rational numbers is not complete, because e.g. 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. == Examples == The space Q of rational numbers, with the standard metric given by the absolute value of the difference, is not complete. Consider for instance the sequence defined by and . This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit ''x'', then necessarily ''x''2 = 2, yet no rational number has this property. However, considered as a sequence of real numbers, it does converge to the irrational number . The open interval , again with the absolute value metric, is not complete either. The sequence defined by ''x''''n'' = is Cauchy, but does not have a limit in the given space. However the closed interval is complete; for example the given sequence does have a limit in this interval and the limit is zero. 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 R''n'', with the usual distance metric. In contrast, infinite-dimensional normed vector spaces may or may not be complete; those that are complete are Banach spaces. The space C 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 of continuous functions on , for it may contain unbounded functions. Instead, with the topology of compact convergence, C can be given the structure of a Fréchet space: a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric. The space Q''p'' of ''p''-adic numbers is complete for any prime number ''p''. This space completes Q with the ''p''-adic metric in the same way that R completes Q with the usual metric. If ''S'' is an arbitrary set, then the set ''S''N of all sequences in ''S'' becomes a complete metric space if we define the distance between the sequences (''x''''n'') and (''y''''n'') to be , where ''N'' is the smallest index for which ''x''''N'' is distinct from ''y''''N'', or 0 if there is no such index. This space is homeomorphic to the product of a countable number of copies of the discrete space ''S''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Complete metric space」の詳細全文を読む スポンサード リンク
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