A generalization of the Banach-Steinhaus theorem for finite part limits

It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence ${\left\{y_n\right\}}_{n=1}^{\mathrm{\infty }}$ of linear continuous functionals in a Fr'echet space converges pointwise to a linear functional $Y,$ $Y\left( x\right) =\lim_{n\rightarrow\infty}\left\langle y_{n},x\right\rangle $forall$x,$then$Y$ is actually continuous. In this article we prove that in a Fr'echet space the continuity of $Y$ still holds if $Y$ is the \emph{finite part} of the limit of $⟨y_n,x⟩$ as $n\to \mathrm{\infty }\mathrm{.}$ We also show that the continuity of finite part limits holds for other classes of topological vector spaces, such as LF-spaces, DFS-spaces, and DFS${}^{\ast }$-spaces, and give examples where it does not hold.