Locally Convex Space is Hausdorff iff induces Hausdorff Topology

Theorem
Let $\struct {X, \mathcal P}$ be a locally convex space.

Let $\tau$ be the standard topology on $\struct {X, \mathcal P}$.

Then $\struct {X, \mathcal P}$ is Hausdorff $\struct {X, \tau}$ is a Hausdorff topological space.

Sufficient Condition
Suppose that $\struct {X, \tau}$ is a Hausdorff.

Let $x \in X$.

If there exists no $y \in X$ with $x \ne y$, then $x = 0$ and $X = \set 0$.

Then there exists no $x \in X$ with $x \ne 0$.

Then $\struct {X, \mathcal P}$ is Hausdorff by vacuous truth.

Otherwise, suppose that $X \ne \set 0$ and there exists $y \in X$ with $x \ne y$.

Since $X$ is a vector space, we then have $y - x \in X$.

Then since $\struct {X, \tau}$ is Hausdorff, there exists $U, V \in \tau$ with $x \in U$ and $y - x \in V$ and $U \cap V = \O$.

From the definition of the standard topology on $\struct {X, \mathcal P}$, there exists $\epsilon > 0$.

Then there exists $\epsilon > 0$ and seminorms $p_1, p_2, \ldots, p_n$ such that:


 * $\set {z \in X : \map {p_k} {z - \paren {y - x} } < \epsilon, \text { for all } k \in \set {1, 2, \ldots, n} } \subseteq V$

Since $x \in U$ we have $x \not \in V$.

So:


 * $x \not \in \set {z \in X : \map {p_k} {z - \paren {y - x} } < \epsilon, \text { for all } k \in \set {1, 2, \ldots, n} }$

Then there exists $k \in \set {1, 2, \ldots, n}$ such that:


 * $\map {p_k} {y - \paren {y - x} } \ge \epsilon > 0$

That is:


 * $\map {p_k} x \ne 0$

So $\struct {X, \mathcal P}$ is Hausdorff.

Necessary Condition
Suppose that $\struct {X, \mathcal P}$ is Hausdorff.

Let $x, y \in X$ have $x \ne y$.

We aim to find $U, V \in \tau$ such that $x \in U$ and $y \in V$ with $U \cap V = \O$.

Since $x \ne y$, we have $y - x \ne 0$.

So from the definition of a Hausdorff locally convex space, there exists $p \in \mathcal P$ such that:


 * $\map p {y - x} \ne 0$

That is:


 * $\map p {y - x} > 0$

since seminorms map onto the non-negative real numbers.

Let:


 * $\epsilon = \map p {y - x}$

Then consider:


 * $U = \set {z \in X : \map p {z - x} < \epsilon/2}$

and:


 * $V = \set {z \in X : \map p {z - y} < \epsilon/2}$

It is clear from the definition of the standard topology on $\struct {X, \mathcal P}$ that $U, V \in \tau$.

From Seminorm Fixes Zero, we have:


 * $\map p 0 = 0$

So we have $x \in U$ and $y \in V$.

We now show that $U \cap V = \O$.

Let $z \in U \cap V$.

Then:


 * $\map p {z - x} < \epsilon/2$

and:


 * $\map p {z - y} < \epsilon/2$

Then, we have:

a contradiction.

So there exists no such $z \in U \cap V$, so $U \cap V = \O$.

Since $x, y \in X$ were arbitrary, we have that $\struct {X, \tau}$ is Hausdorff.