习题 1, § 4

令$E$为向量空间，且$A$为$E$的凸对称凸子集。 令$\mathscr{T}, \mathscr{T}'$为$E$上的两个局部凸拓扑，$\mathscr{U}, \mathscr{U}'$为$\mathscr{T}, \mathscr{T}'$在$E$上定义的一致结构。 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 $\mathscr{T}'$.