![]() This space completes Q with the p-adic metric in the same way that R completes Q with the usual metric. The space Q p of p-adic numbers is complete for any prime number p. 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. 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. ![]() 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. In contrast, infinite-dimensional normed vector spaces may or may not be complete those that are complete are Banach spaces. 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. However the closed interval is complete the given sequence does have a limit in this interval and the limit is zero. ![]() The sequence defined by x n = 1⁄ n is Cauchy, but does not have a limit in the given space. The open interval (0, 1), again with the absolute value metric, is not complete either. However, considered as a sequence of real numbers, it does converge to the irrational number. 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. Consider for instance the sequence defined by and. The space Q of rational numbers, with the standard metric given by the absolute value, is not complete. ![]()
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