Exercise 1, § 4

Let $E$ be a vector space and $A$ a convex symmetric convex subset of $E$. Let $\mathscr{T}, \mathscr{T}'$ be two locally convex topologies on $E$ and $\mathscr{U}, \mathscr{U}'$ be the uniform structures defined by $\mathscr{T}, \mathscr{T}'$ on $E$. In order that the uniform structure induced on $A$ by $\mathscr{U}'$ should be finer than that induced by $\mathscr{U}$, it is necessary and sufficient that every neighbourhood of 0 for the topology induced on $A$ by $\mathscr{T}$ should be a neighbourhood of 0 for the topology induced on $A$ by $\mathscr{T}'$.