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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.

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Bulletin of the Malaysian Mathematical Sciences Society

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