Complete Metric Space - Examples

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 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 x2 = 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 (0, 1), again with the absolute value metric, is not complete either. The sequence defined by x_n = 1⁄_n is Cauchy, but does not have a limit in the given space. However the closed interval is complete; 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 Rn, 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(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 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 SN 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 1⁄_N, 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.