Category:Cauchy Sequences in Topological Vector Spaces

This category contains results about Cauchy Sequences in Topological Vector Spaces.
Definitions specific to this category can be found in Definitions/Cauchy Sequences in Topological Vector Spaces.

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 $n, m \ge N$.