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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A Steinhaus type theorem


Author: P. N. Natarajan
Journal: Proc. Amer. Math. Soc. 99 (1987), 559-562
MSC: Primary 40C05; Secondary 40D25
DOI: https://doi.org/10.1090/S0002-9939-1987-0875397-2
MathSciNet review: 875397
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Abstract: The sequence space $ {\Lambda _r},r \geq 1$ being a fixed integer, is defined as

$\displaystyle {\Lambda _r} = \left\{ {x = \left\{ {{x_k}} \right\} \in {l_\inft... ...ts ,\left\vert {{x_{k + r}} - {x_k}} \right\vert \to 0,k \to \infty } \right\},$

where $ K$ is a complete, nontrivially valued field and $ {l_\infty }$ is the space of bounded sequences with entries in $ K$. In this paper, it is proved that given a regular matrix $ A = ({a_{nk}}),{a_{nk}} \in K = {\mathbf{R}}$ or $ {\mathbf{C}}$, there exists a sequence in $ {\Lambda _r} - \cup _{i = 1}^{r - 1}{\Lambda _i}$ which is not $ A$-summable. This is an improvement of the well-known Steinhaus theorem. It is, however, shown that this result fails to hold when $ K$ is a complete, nontrivially valued, nonarchimedean field, whereas it is known that the Steinhaus theorem continues to hold.

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DOI: https://doi.org/10.1090/S0002-9939-1987-0875397-2
Article copyright: © Copyright 1987 American Mathematical Society