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

Document Type : Dedicated to Prof. A. T.-M. Lau

Authors

Département de mathématiques et de statistique, Université Laval, Québec City (Québec), G1V 0A6, Canada.

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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References

  1. S. Banach and H. Steinhaus, Sur le principe de la condensation de singularités, Fund. Math. 9 (1927) 50–61.
  2. S. Ghara, J. Mashreghi and T. Ransford, Summability and duality, preprint, 2022.
  3. W. Rudin, Functional Analysis, Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
Journal of the Iranian Mathematical Society
Volume 3, Issue 2
July 2022
Pages 43-47

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