How are the assumptions used in the proof of Bourbaki-Alaoglu Theorem?

Question

This is a follow up question to a previous one.

In the proof of the following theorem, where are the assumptions "Hausdorff" and "locally convex" used?

Answer 1

There is a theorem says that

If $X$ is a Hausdorff locally convex space, and $x,y$ are two distinct points in $X$, there exists $\lambda\in X^*$ such that $\varphi(x)\not=\varphi(y)$.

The assumption "Hausdorff locally convex space" is used to make $(X^*,X)$ as a dual pair.

In general, one has the following statement:

If $X$ and $Y$ are in duality, then $V^\circ$ is compact in the topology $\sigma(Y,X)$ whenever $V$ is a neighborhood of zero in a locally convex topology compatible with the duality.