Corollary 2

— Suppose that the topology $\mathscr{T}$ of a topological vector space $E$ over $K$ is defined by a directed set of semi-norms $\Gamma$; for each semi-norm $p \in \Gamma$, let $E_p$ be the space obtained from $E$ using the topology defined by $p$. The set $E'$ of linear forms on $E$ that are continuous for $\mathscr{T}$ is the union of the sets $E'_p$, where $E'_p$ is the set of continuous linear forms in $E_p$ ($p \in \Gamma$).