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A Steinhaus type theorem

Author: P. N. Natarajan
Journal: Proc. Amer. Math. Soc. 99 (1987), 559-562
MSC: Primary 40C05; Secondary 40D25
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.

References [Enhancements On Off] (What's this?)

  • [1] G. Bachman, Introduction to $ p$-adic numbers and valuation theory, Academic Press, New York, 1964. MR 0169847 (30:90)
  • [2] L. Narici, E. Beckenstein, and G. Bachman, Functional analysis and valuation theory, Dekker, New York, 1971. MR 0361697 (50:14142)
  • [3] P. N. Natarajan, The Steinhaus theorem for Toeplitz matrices in non-archimedean fields, Comment. Math. Prace Mat. 20 (1978), 417-422. MR 519377 (80f:40007)
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Article copyright: © Copyright 1987 American Mathematical Society

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