Vector series whose lacunary subseries converge

Drewnowski, Lech, and Labuda, Iwo. "Vector series whose lacunary subseries converge." Studia Mathematica 138.1 (2000): 53-80. <http://eudml.org/doc/216690>.

@article{Drewnowski2000,
abstract = {The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its zero-density subseries converge. A series $∑_n x_n$ in a topological vector space X is called ℒ-convergent if each of its lacunary subseries $∑_k x_\{n_k\}$ (i.e. those with $n_\{k+1\} - n_k → ∞$) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every ℒ-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence Property, or ZCP, is defined similarly though of lesser importance here. It is shown that for every ℒ-convergent series the set of all its finite sums is metrically bounded; however, it need not be topologically bounded. Next, a space with the LCP contains no copy of the space $c_0$. The converse holds for Banach spaces and, more generally, sequentially complete locally pseudoconvex spaces. However, an F-lattice of measurable functions is constructed that has both the Lebesgue and Levi properties, and thus contains no copy of $c_0$, and, nonetheless, lacks the LCP. The main (and most difficult) result of the paper is that if a Banach space E contains no copy of $c_0$ and λ is a finite measure, then the Bochner space $L_0$ (λ,e) has the LCP. From this, with the help of some Orlicz-Pettis type theorems proved earlier by the authors, the LCP is deduced for a vast class of spaces of (scalar and vector) measurable functions that have the Lebesgue type property and are “metrically-boundedly sequentially closed” in the containing $L_0$ space. Analogous results about the convergence of ℒ-convergent positive series in topological Riesz spaces are also obtained. Finally, while the LCP implies the ZCP trivially, an example is given that the converse is false, in general.},
author = {Drewnowski, Lech, Labuda, Iwo},
journal = {Studia Mathematica},
keywords = {subseries convergence; lacunary subseries; zero-density subseries; lacunary convergence property; topological Riesz space of measurable functions; topological vector space of Bochner measurable functions; Lebesgue property; Levi property; copy of $c_0$; Banach space; Bochner space},
language = {eng},
number = {1},
pages = {53-80},
title = {Vector series whose lacunary subseries converge},
url = {http://eudml.org/doc/216690},
volume = {138},
year = {2000},
}

TY - JOUR
AU - Drewnowski, Lech
AU - Labuda, Iwo
TI - Vector series whose lacunary subseries converge
JO - Studia Mathematica
PY - 2000
VL - 138
IS - 1
SP - 53
EP - 80
AB - The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its zero-density subseries converge. A series $∑_n x_n$ in a topological vector space X is called ℒ-convergent if each of its lacunary subseries $∑_k x_{n_k}$ (i.e. those with $n_{k+1} - n_k → ∞$) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every ℒ-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence Property, or ZCP, is defined similarly though of lesser importance here. It is shown that for every ℒ-convergent series the set of all its finite sums is metrically bounded; however, it need not be topologically bounded. Next, a space with the LCP contains no copy of the space $c_0$. The converse holds for Banach spaces and, more generally, sequentially complete locally pseudoconvex spaces. However, an F-lattice of measurable functions is constructed that has both the Lebesgue and Levi properties, and thus contains no copy of $c_0$, and, nonetheless, lacks the LCP. The main (and most difficult) result of the paper is that if a Banach space E contains no copy of $c_0$ and λ is a finite measure, then the Bochner space $L_0$ (λ,e) has the LCP. From this, with the help of some Orlicz-Pettis type theorems proved earlier by the authors, the LCP is deduced for a vast class of spaces of (scalar and vector) measurable functions that have the Lebesgue type property and are “metrically-boundedly sequentially closed” in the containing $L_0$ space. Analogous results about the convergence of ℒ-convergent positive series in topological Riesz spaces are also obtained. Finally, while the LCP implies the ZCP trivially, an example is given that the converse is false, in general.
LA - eng
KW - subseries convergence; lacunary subseries; zero-density subseries; lacunary convergence property; topological Riesz space of measurable functions; topological vector space of Bochner measurable functions; Lebesgue property; Levi property; copy of $c_0$; Banach space; Bochner space
UR - http://eudml.org/doc/216690
ER -