Arrangements of series preserving their convergence or boundedness

Lech Drewnowski. "Arrangements of series preserving their convergence or boundedness." Commentationes Mathematicae 47.1 (2007): null. <http://eudml.org/doc/292206>.

@article{LechDrewnowski2007,
abstract = {For a map $\rho $ of $\mathbb \{N\}$ into itself, consider the induced transformation $\sum _\{n\} x_n \mapsto \sum _\{n\} x_\{\rho _(n)\}$ 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 ρ which reduces to those found by Agnew and Pleasants (in the case of permutations) and Wituła (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.},
author = {Lech Drewnowski},
journal = {Commentationes Mathematicae},
keywords = {Convergent series; bounded series; permutations of series; arrangements of series; topological vector space; spaces of bounded or convergent series},
language = {eng},
number = {1},
pages = {null},
title = {Arrangements of series preserving their convergence or boundedness},
url = {http://eudml.org/doc/292206},
volume = {47},
year = {2007},
}

TY - JOUR
AU - Lech Drewnowski
TI - Arrangements of series preserving their convergence or boundedness
JO - Commentationes Mathematicae
PY - 2007
VL - 47
IS - 1
SP - null
AB - For a map $\rho $ of $\mathbb {N}$ into itself, consider the induced transformation $\sum _{n} x_n \mapsto \sum _{n} x_{\rho _(n)}$ 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 ρ which reduces to those found by Agnew and Pleasants (in the case of permutations) and Wituła (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.
LA - eng
KW - Convergent series; bounded series; permutations of series; arrangements of series; topological vector space; spaces of bounded or convergent series
UR - http://eudml.org/doc/292206
ER -