Regard Schwartz space as a subspace of $L^2$ , then can it be written as a countable union of nowhere dense subset in $L^2$?

Question

Let $\{T_n\}$ be a sequence of functional $T_n: L^2 \to C$ ; $S$ denote the schwartz space . If we can show that $$\lim_n T_n(g)$$ Converges for every $g \in S$ and $S$ is of second category (which means $S$ can not be written as a countable union of nowhere dense set) , then by the uniform boundness principle , we can conclude that $\{T_n\}$ are uniform bounded .

So , can Schwartz space be written as a countable union of nowhere dense set ?