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arXiv · 2302.06722

Asymptotically equicontinuous sequences of operators and a Banach-Steinhaus type theorem

Javad Mashreghi·Thomas Ransford
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Abstract

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.

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Javad Mashreghi, Thomas Ransford. 2023-02-13. Asymptotically equicontinuous sequences of operators and a Banach-Steinhaus type theorem. https://arxiv.org/abs/2302.06722

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