Assume we have a locally convex topology $\tau$ induced by semi-norms $\mathcal P$, on some real vectorspace $E$. Let $\sigma$ be the locally convex topology induced by the semi-norms $$ \mathcal Q := \{|f| : f \text{ is $\tau$-continous and linear} \} $$ Let $A \subset E$ be convex. Is is true that $A$ is $\tau$ closed iff $A$ is $\sigma$ closed.
If $A$ is $\sigma$ closed, the claim is easy to prove, but I am wondering if the opposite is true.
Yes, it's true. That is a consequence of the Hahn-Banach theorem(s).
If $A$ is $\tau$-closed and convex, and $x \in E\setminus A$, there exists a $\tau$-continuous linear functional $f_A$ such that
$$S := \sup \{ f_A(a) : a \in A\} < f_A(x).$$
Then
$$\left\{ y\in E : \lvert f_A(y-x)\rvert < f_A(x) - S\right\}$$
is a $\sigma$-neighbourhood of $x$ that doesn't intersect $A$.