Let $\mathcal{P}$ be a directed family of seminorms on a topological vector space $X$ and let $\tau_{\mathcal{P}}$ be the locally convex topology on $X$ generated by the family $\mathcal{P}$ (hence, $\tau_{\mathcal{P}}$ is the weakest locally convex topology on $X$ such that each $p\in \mathcal{P}$ is continuous on $X$).
If we take any $\tau_{\mathcal{P}}$-continuous seminorm $q$ on $X$ with $q\not\in\mathcal{P}$, does there exist a relation between $q$ and the family $\mathcal{P}$ ? In particular, does there exist a $p\in \mathcal{P}$ and a constant $c>0$ such that $q(x)\leq c p(x)$ for all $x\in X$ ?
If $q$ is $\tau_\mathcal P$-continuous its unit ball $B_q(0,1)=\lbrace x\in X: q(x)<1\rbrace$ is $\tau_\mathcal P$-open and contains $0$. Hence there are $p\in \mathcal P$ and $\varepsilon>0$ with $B_p(0,\varepsilon)\subseteq B_q(0,1)$ and the homogeneity of the seminorms implies $q\le cp$ for $c=1/\varepsilon$.