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Wyszukujesz frazę "Lebesgue measurable" wg kryterium: Temat


Wyświetlanie 1-1 z 1
Tytuł:
Vector series whose lacunary subseries converge
Autorzy:
Drewnowski, Lech
Labuda, Iwo
Powiązania:
https://bibliotekanauki.pl/articles/1206244.pdf
Data publikacji:
2000
Wydawca:
Polska Akademia Nauk. Instytut Matematyczny PAN
Tematy:
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 $c_0$
Opis:
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.
Źródło:
Studia Mathematica; 2000, 138, 1; 53-80
0039-3223
Pojawia się w:
Studia Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
    Wyświetlanie 1-1 z 1

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