Let $X$ be a real vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each $p\in P$ continuous with respect to $\sigma(X,P)$. It can be shown that $\sigma(X,P)$ is a Hausdorff topology compatible with the vector space structure of $X$, see for instance this and this one.
Question. How do we construct a base of convex open neighborhoods of $0$ in $X$ for the the weak topology $\sigma(X,P)$?
Each seminorm $p$ gives you convex open sets $\{p<t\}$, for various positive numbers $t$. Take all of these sets, and you'll have a sub-basis of neighborhoods. (From which a basis is obtained by taking finite intersections).