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

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 Fréchet space converges pointwise to a linear functional $Y,$ $Y\left(x\right)={\lim }_{n\to \infty }⟨y_n,x⟩$ for all $x,$ then $Y$ is actually continuous. In this article we prove that in a Fréchet space the continuity of $Y$ still holds if $Y$ is the \emph{finite part} of the limit of $⟨y_n,x⟩$ as $n\to \infty .$ 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.

13 pages

References in corpus (1)

• Generalized second-order partial derivatives of 1/r

Cited by in corpus (1)

• Distributions in spaces with thick submanifolds