#### Title

A Generalization of the Banach-Steinhaus Theorem for Finite Part Limits

#### Document Type

Article

#### Publication Date

4-1-2017

#### Abstract

It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence {yn}n=1∞ of linear continuous functionals in a Fréchet space converges pointwise to a linear functional Y, Y(x) = lim n → ∞〈yn, 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 finite part of the limit of 〈yn, x〉 as n→ ∞. 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∗-spaces and give examples where it does not hold.

#### Publication Source (Journal or Book title)

Bulletin of the Malaysian Mathematical Sciences Society

#### First Page

907

#### Last Page

918

#### Recommended Citation

Estrada, R., & Vindas, J.
(2017). A Generalization of the Banach-Steinhaus Theorem for Finite Part Limits.* Bulletin of the Malaysian Mathematical Sciences Society**, 40* (2), 907-918.
https://doi.org/10.1007/s40840-017-0450-7