Example

— Let $(a_i)_{i \in I}$ be a basis of the vector space $E$ and consider the canonical topology (I, p. 2, Example 5) on each line $Ra_i$; the direct sum of these topologies is the finest locally convex topology on $E$ (II, p. 26); in fact, if $V$ is an absorbent, symmetric, convex set in $E$, then $V_i = V \cap Ra_i$ is a neighbourhood of 0 in $Ra_i$ and $V$ clearly contains the convex envelope $\Gamma((V_i))$.