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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A Steinhaus type theorem
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by P. N. Natarajan
Proc. Amer. Math. Soc. 99 (1987), 559-562
DOI: https://doi.org/10.1090/S0002-9939-1987-0875397-2

Abstract:

The sequence space ${\Lambda _r},r \geq 1$ being a fixed integer, is defined as \[ {\Lambda _r} = \left \{ {x = \left \{ {{x_k}} \right \} \in {l_\infty },{x_k} \in K,k = 0,1,2, \ldots ,\left | {{x_{k + r}} - {x_k}} \right | \to 0,k \to \infty } \right \},\] 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
  • George Bachman, Introduction to $p$-adic numbers and valuation theory, Academic Press, New York-London, 1964. MR 0169847
  • Lawrence Narici, Edward Beckenstein, and George Bachman, Functional analysis and valuation theory, Pure and Applied Mathematics, Vol. 5, Marcel Dekker, Inc., New York, 1971. MR 0361697
  • P. N. Natarajan, The Steinhaus theorem for Toeplitz matrices in non-Archimedean fields, Comment. Math. Prace Mat. 20 (1977/78), no. 2, 417–422. MR 519377
  • K. Zeller and W. Beekmann, Theorie der Limitierungsverfahren, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 15, Springer-Verlag, Berlin-New York, 1970 (German). Zweite, erweiterte und verbesserte Auflage. MR 0264267
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Bibliographic Information
  • © Copyright 1987 American Mathematical Society
  • 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