Proof. $(1) \Rightarrow (2)$: Let $U \in \cn(0)$ be convex. By Proposition 8.1.11, there exists $V \in \cn(0)$ circled such that $V + V \subset U$. Let $W = \bracs{tx + (1 - t)y|x, y \in V}$ be the convex hull of $V$, then $W \subset U$ is convex and circled.

$(2) \Rightarrow (3)$: For each $V \in \cn(0)$ convex, circled, and radial, let $[\cdot]_{V}: E \to [0, \infty)$ be its gauge, then $[\cdot]_{V}$ is a seminorm. For each $x, y \in X$ and $r > 0$, $[x - y]_{V} < r$ if and only if $x - y \in rV$. Thus the uniformity induced by $[\cdot]_{V}$ corresponds to the uniformity generated by $\bracs{U_{rV}| r > 0}$, where $U_{V} = \bracs{(x, y) \in E|x - y \in V}$. Since this holds for all $U \in \cn(0)$, the topology of $E$ and the topology induced by $\bracs{[\cdot]_V| V \in \cn(0), \text{ convex, circled, radial}}$ coincide.

$(3) \Rightarrow (1)$: For each $i \in I$ and $r > 0$, $\bracs{x \in E| [x]_i < r}$ is convex.$\square$