Definition:Cauchy Sequence/Topological Vector Space

Definition

Let $\struct {X, \tau}$ be a topological vector space.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.

We say that $\sequence {x_n}_{n \mathop \in \N}$ is Cauchy if and only if:

for each open neighborhood $V$ of ${\mathbf 0}_X$ there exists $N \in \N$ such that:

$x_n - x_m \in V$ for each $n, m \ge N$.

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.25$: Cauchy sequences