Arrangements of series preserving their convergence or boundedness

• Autorzy: Drewnowski L.
• Języki publikacji: EN

Abstrakty

EN

For a map p of N into itself, consider the induced transformation [...] of series in a topological vector space. Then such properties of this transformation as sending convergent series to convergent series, or convergent series to bounded series, or bounded series to bounded series (and a few more) are mutually equivalent. Moreover, they are equivalent to an intrinsic property of p which reduces to those found by Agnew and Pleasants (in the case of permutations) and Witula (in the general case) as necessary and sufficient conditions for the above transformation to preserve convergence of scalar series. In the paper, the scalar case is treated first using simple Banach space methods, and then the result is easily extended to the general setting.

Słowa kluczowe

EN

convergent series; bounded series; permutations of series; arrangements of series; topological vector space; spaces of bounded or convergent series

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
• Rocznik: 2007
• Tom: Vol. 47, [Z] 1
• Strony: 57--75
• Opis fizyczny: bibliogr. 10 poz.

Twórcy

autor

Drewnowski L.

• Faculty of Mathematics and Computer Sciences, A.Mickiewicz University Umultowska 87, 61-614 Poznań, Poland,

Bibliografia

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• [2] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience, New York 1958.
• [3] H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart 1981.
• [4] P. Kostyrko, On convergence preserving transformations of infinite series, Math. Slovaca 46 (1996), 239-243.
• [5] P. A. B. Pleasants, Rearrangements that preserve convergence, J. London Math. Soc. (2) 15 (1977), 134-142, and Addendum, ibid. 18 (1978), 576.
• [6] M. Ali Sarigol, A short proof of Levi's theorem on rearrangements of convergent series, Doga-Mat. 16 (1992), 201-205.
• [7] M. Ali Sarigol, Rearrangements of bounded variation sequences, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), 373-376.
• [8] P. Schaefer, Sum-preserving rearrangement of convergent series, Amer. Math. Monthly 88 (1981), 33-40.
• [9] A. Wilansky, Summability through Functional Analysis, Notas de Matematica no. 85, North-Holland, Amsterdam, New York and Oxford 1984.
• [10] R. Witula, Convergence-preserving functions, Nieuw.-Arch.-Wisk. (4) 13 (1995), 31-35.