For the following problem concerning functional analysis I cannot seem to find a solution. It states as follows.
Let $(E, \tau)$ be a locally convex vector space and $S \subseteq E$ with $S \ni 0$. Show that $S$ is closed and convex iff there is a subset $T \subseteq E^*$ such that $$S = \bigcap_{f \in T}\{\text{Re}\, f(x) \leq 1\}.$$
The "$\Rightarrow$"-direction seems to somehow follow from the separation theorem version of the Hahn-Banach theorem as I read it in Rudin's "Functional Analysis", p. 59, Theorem 3.4. Although I do not really know how to apply it here?
For the "$\Leftarrow$"-direction, the closedness of $S$ as intersection of closed sets seems clear to me. For the convexity again I am not so sure but I suppose this is due to the space being locally convex. Since $0 \in S$ we can find some absolutely convex base around 0.
What is the correct line of argument here?