I was given the following theorem by my professor:
Let $A$ be a locally convex space. Then $A$ is Hausdorff if and only if, for every seminorm $p$, we have that $p(x)=0$ implies that $x=0$.
If I am not mistaken, what this actually says, is that every locally convex space is Hausdorff if and only if every seminorm is also a norm (i.e. if the locally convex space is also a normed space).
Am I understanding this statement correctly? Can anyone please provide me with a nice reference of this result?
This statement is false, the weak topology is Haussdorff but doesn't have the given property.
What is true is that $A$ is Haussdorff if and only if for each $x \neq 0$ there exists a $p$ such that $p(x) \neq 0$.