Corollary 1

— Suppose that $\mathscr{T}, \mathscr{T}'$ are two topologies on a vector space $E$ over $K$ defined, respectively, by two sets of semi-norms $\Gamma$ and $\Gamma'$. $\mathscr{T}$ is finer than $\mathscr{T}'$ if, and only if, for every semi-norm $q \in \Gamma'$, there exists a finite family $(p_i)_{1 \leq i \leq n}$ of semi-norms belonging to $\Gamma$ and a number $a > 0$ such that, for all $x \in E$, we have $q(x) \leq a \cdot \sup_{1 \leq i \leq n} p_i(x)$.

In fact this shows that the identity mapping of $E$ with topology $\mathscr{T}$, on $E$ with topology $\mathscr{T}'$, is continuous.