Exercise 23, § 5

Let $E$ be a topological vector space, and let $\mathscr{T}$ be the locally convex topology on $E$ that is the finest of all those that are coarser than the given topology $\mathscr{T}_0$ on $E$. If $F$ is a locally convex space, then the continuous linear mappings of $E$ in $F$ are the same for $\mathscr{T}_0$ as for $\mathscr{T}$. There exists a continuous linear form on $E$ that is distinct from the null form if, and only if, there exists a neighbourhood of 0 for $\mathscr{T}_0$ whose convex envelope is not everywhere dense (for $\mathscr{T}_0$) (cf. I, p. 25, exerc. 4).