%0 Journal Article %T Asymptotically equicontinuous sequences of operators and a Banach--Steinhaus type theorem %J Journal of the Iranian Mathematical Society %I Iranian Mathematical Society %Z 2717-1612 %A Mashreghi, J. %A Ransford, T. %D 2022 %\ 07/01/2022 %V 3 %N 2 %P 43-47 %! Asymptotically equicontinuous sequences of operators and a Banach--Steinhaus type theorem %K Banach-Steinhaus theorem %K dense %K equicontinuous %R 10.30504/jims.2022.361848.1073 %X We introduce the notion of an asymptotically equicontinuous sequence of linear operators, and use it to prove the following result. If $X,Y$ are topological vector spaces, if $T_n,T:X\to Y$ are continuous linear maps, and if $D$ is a dense subset of $X$, then the following statements are equivalent: $(i) ~T_nx\to Tx$ for all $x\in X$, and $(ii) ~T_n x\to Tx$ for all $x\in D$ and the sequence $(T_n)$ is asymptotically equicontinuous. %U https://jims.ims.ir/article_158793_4504b652d02694be4d859572ff4f36e6.pdf