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    Wyświetlanie 1-3 z 3
    Tytuł:
    Arrangements of series preserving their convergence or boundedness
    Autorzy:
    Drewnowski, Lech
    Powiązania:
    https://bibliotekanauki.pl/articles/745881.pdf
    Data publikacji:
    2007
    Wydawca:
    Polskie Towarzystwo Matematyczne
    Tematy:
    Convergent series
    bounded series
    permutations of series
    arrangements of series
    topological vector space
    spaces of bounded or convergent series
    Opis:
    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.
    Źródło:
    Commentationes Mathematicae; 2007, 47, 1
    0373-8299
    Pojawia się w:
    Commentationes Mathematicae
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    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
    Tytuł:
    Nonseparability of the quotient space cabv(∑,m;X)/L¹(m;X) for Banach spaces X without the Radon-Nikodym property
    Autorzy:
    Drewnowski, Lech
    Powiązania:
    https://bibliotekanauki.pl/articles/1292918.pdf
    Data publikacji:
    1993
    Wydawca:
    Polska Akademia Nauk. Instytut Matematyczny PAN
    Tematy:
    Banach space
    spaces of vector measures
    Bochner integrable functions
    Radon-Nikodym property
    nonseparable quotient space
    Opis:
    It is shown that if (S,∑,m) is an atomless finite measure space and X is a Banach space without the Radon-Nikodym property, then the quotient space cabv(∑,m;X)/L¹(m;X) is nonseparable.
    Źródło:
    Studia Mathematica; 1993, 104, 2; 125-132
    0039-3223
    Pojawia się w:
    Studia Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-3 z 3

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