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.
Mashreghi, J., & Ransford, T. (2022). Asymptotically equicontinuous sequences of operators and a Banach--Steinhaus type theorem. Journal of the Iranian Mathematical Society, (), -. doi: 10.30504/jims.2022.361848.1073
MLA
J. Mashreghi; T. Ransford. "Asymptotically equicontinuous sequences of operators and a Banach--Steinhaus type theorem". Journal of the Iranian Mathematical Society, , , 2022, -. doi: 10.30504/jims.2022.361848.1073
HARVARD
Mashreghi, J., Ransford, T. (2022). 'Asymptotically equicontinuous sequences of operators and a Banach--Steinhaus type theorem', Journal of the Iranian Mathematical Society, (), pp. -. doi: 10.30504/jims.2022.361848.1073
VANCOUVER
Mashreghi, J., Ransford, T. Asymptotically equicontinuous sequences of operators and a Banach--Steinhaus type theorem. Journal of the Iranian Mathematical Society, 2022; (): -. doi: 10.30504/jims.2022.361848.1073