Olivier’s theorem: ideal convergence, algebrability and Borel classification

The classical Olivier’s theorem says that for any nonincreasing summable sequence ( a ( n )) the sequence ( na ( n )) tends to zero. This result was generalized by many authors. We propose its further generalization which implies known results. Next we consider the subset AOS of ℓ 1 consisting of se...

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Bibliographic Details
Published in:Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Físicas y Naturales. Serie A, Matemáticas, 2021-10, Vol.115 (4), Article 200
Main Authors: Bartoszewicz, Artur, Gła̧b, Szymon, Widz, Agnieszka
Format: Article
Language:English
Subjects:
Applications of Mathematics
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Original Paper
Theorems
Theoretical
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Summary:The classical Olivier’s theorem says that for any nonincreasing summable sequence ( a ( n )) the sequence ( na ( n )) tends to zero. This result was generalized by many authors. We propose its further generalization which implies known results. Next we consider the subset AOS of ℓ 1 consisting of sequences for which the assertion of Olivier’s theorem is false. We study how large and good algebraic structures are contained in AOS and its subsets; this kind of study is known as lineability. Finally we show that AOS is a residual G δ σ but not an F σ δ -set .
ISSN:1578-7303
1579-1505
DOI:10.1007/s13398-021-01143-y