Equivalence of Definitions of Hausdorff Topological Vector Space

Theorem
Let $K$ be a topological field.

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

Proof
From Topological Vector Space is Hausdorff iff T1, $\struct {X, \tau}$ is Hausdorff it is $T_1$ space.

From Definition 2 of a $T_1$ space, it follows that $\struct {X, \tau}$ is Hausdorff :
 * for each $x \in X$, $\set x$ is closed in $\struct {X, \tau}$.