Let $(X,\mathcal{P}_X)$ and $(Y,\mathcal{P}_Y)$ be locally convex topological vector spaces with topologies induced by the families of continuous seminorms $\mathcal{P}_X$ and $\mathcal{P}_Y$ respectively and let $A:X\to Y$ be a linear map.
I want to show that $A:X\to Y$ is continuous $\iff\forall q\in\mathcal{P}_Y$ there exists $p\in\mathcal{P}_X$ such that $q(A(x))\le p(x)$ for all $x\in X$.
I wanted to first show the $\Longleftarrow$ direction.
Via the pre-image
First I tried to do it by claiming that it suffices to show that $A$ is continuous at 0 in $X$. To this end, let $B_q(0,r)$, with $q\in\mathcal{P}_Y$ and $r>0$ be an open neighbourhood of $0$ in $Y$. Then we must show that the pre-image $A^{-1}(B_q(0,r))$ is open in $X$. That is, $A^{-1}(B_q(0,r))=\{x\in X:T(x)\in B_q(0,r)\}$...
However, this seemed a dead-end and I couldn't see how to relate it with the inequality.
Showing A is bounded
Next I considered that $A:X\to Y$ is continuous if and only if $A:X\to Y$ is bounded. And, in a locally convex topological space, $U\subset X$ is bounded if and only if every $p\in\mathcal{P}_X$ is bounded on $U$. Thus, since the topology on $Y$ is induced by $\mathcal{P}_Y$, the map $A:X\to Y$ is bounded if and only if for all $q\in\mathcal{P}_Y$ there exists $p\in\mathcal{P}_X$ such that $q(A(x))\le p(x)$...which would in fact prove the if and only if statement, however, I am not sure if this argument is complete.
I also thought of proving it via some definition of sequential continuity, e.g. $A$ continuous if and only if $x_n\to 0\implies Ax_n\to 0$, but it seems a little tricky to formulate in this seminorm setting and I don't know if it would be fruitful.