Complete metric space

In mathematics, completeness is a property ascribed to a metric space in which every Cauchy sequence in that space is convergent. In other words, every Cauchy sequence in the metric space tends in the limit to a point which is again an element of that space. Hence the metric space is, in a sense, "complete."

Formal definition

Let X be a metric space with metric d. Then X is complete if for every Cauchy sequence $x_{1},x_{2},\ldots \in X$ there is an associated element $x\in X$ such that $\mathop {\lim } _{n\rightarrow \infty }d(x_{n},x)=0$.

See also

Banach space

Hilbert space