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

TL;DR

The paper introduces asymptotically equicontinuous sequences of operators and proves a Banach-Steinhaus type theorem relating pointwise convergence and asymptotic equicontinuity in topological vector spaces.

Contribution

It defines asymptotic equicontinuity for operator sequences and establishes a new Banach-Steinhaus type result involving dense subsets and convergence.

Findings

Characterization of convergence via dense subsets and asymptotic equicontinuity

Introduction of asymptotically equicontinuous sequences of operators

Extension of Banach-Steinhaus theorem to topological vector spaces

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.