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

TL;DR

This paper extends the Banach-Steinhaus theorem to finite part limits in Fréchet spaces and other topological vector spaces, establishing conditions under which these limits are continuous functionals.

Contribution

It generalizes the Banach-Steinhaus theorem to finite part limits in various topological vector spaces, including Fréchet, LF, DFS, and DFS* spaces.

Findings

Finite part limits are continuous in Fréchet spaces.

Continuity of finite part limits extends to LF, DFS, DFS* spaces.

Examples show cases where finite part limits are not continuous.

Abstract

It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence $\left\{y_{n}\right\}_{n=1}^{\infty}$ 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 $ for all $x,$ 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 $\left\langle y_{n},x\right\rangle $ as $n\rightarrow\infty.$ 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.