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, 3(2), 43–47. https://doi.org/10.30504/jims.2022.361848.1073
MLA
Mashreghi, J., & Ransford, T. "Asymptotically equicontinuous sequences of operators and a Banach--Steinhaus type theorem", Journal of the Iranian Mathematical Society, 3, 2, 2022, 43–47. 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, 3(2), pp. 43–47. doi: 10.30504/jims.2022.361848.1073
CHICAGO
Mashreghi, J. & Ransford, T., "Asymptotically equicontinuous sequences of operators and a Banach--Steinhaus type theorem." Journal of the Iranian Mathematical Society, 3 2 (2022): 43–47, doi: 10.30504/jims.2022.361848.1073
VANCOUVER
Mashreghi J., Ransford T. Asymptotically equicontinuous sequences of operators and a Banach--Steinhaus type theorem. JIMS. 2022;3(2):43–47. doi: 10.30504/jims.2022.361848.1073
