Category:Definitions/Hausdorff Topological Vector Spaces

This category contains definitions related to Hausdorff Topological Vector Spaces.
Related results can be found in Category:Hausdorff Topological Vector Spaces.

Let $K$ 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 $\struct {X, \tau}$.