Definition:Hausdorff Topological Vector Space

Definition

Let $\struct {K, +_K, \circ_K, \tau_K}$ be a topological field.

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

Definition 1

We say that $\struct {X, \tau}$ is a Hausdorff topological vector space if and only if it is Hausdorff as a topological space.

Definition 2

We say that $\struct {X, \tau}$ is a Hausdorff topological vector space if and only if:

for each $x \in X$, the singleton $\set x$ is closed in $\struct {X, \tau}$.

Also see

• Equivalence of Definitions of Hausdorff Topological Vector Space

• Results about Hausdorff topological vector spaces can be found here.