Cauchy sequence
In topological vector spaces

There is also a concept of Cauchy sequence for a topological vector space X {\displaystyle X}: Pick a local base B {\displaystyle B} for X {\displaystyle X} about 0; then (x k {\displaystyle x_{k}}) is a Cauchy sequence if for each member V ∈ B, {\displaystyle V\in B,} there is some number N {\displaystyle N} such that whenever n, m > N, x n − x m {\displaystyle n,m>N,x_{n}-x_{m}} is an element of V. {\displaystyle V.} If the topology of X {\displaystyle X} is compatible with a translation-invariant metric d, {\displaystyle d,} the two definitions agree.
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