# Vector series whose lacunary subseries converge

Studia Mathematica (2000)

- Volume: 138, Issue: 1, page 53-80
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topDrewnowski, 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 -

## References

top- [Ag] R. P. Agnew, Subseries of series which are not absolutely convergent, Bull. Amer. Math. Soc. 53 (1947), 118-120. Zbl0037.04704
- [AB] C. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces, Academic Press, 1978. Zbl0402.46005
- [Au] H. Auerbach, Über die Vorzeichenverteilung in unendlichen Reihen, Studia Math. 2 (1930), 228-230. Zbl56.0200.02
- [B] S. Banach, Théorie des opérations linéaires, Monografje matematyczne, Warszawa, 1932.
- [BDV] J. Batt, P. Dierolf and J. Vogt, Summable sequences and topological properties of ${m}_{0}\left(I\right)$, Arch. Math. (Basel) 28 (1977), 86-90. Zbl0362.46007
- [BP] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. Zbl0084.09805
- [DDD] P. Dierolf, S. Dierolf and L. Drewnowski, Remarks and examples concerning unordered Baire-like and ultrabarrelled spaces, Colloq. Math. 39 (1978), 109-116. Zbl0386.46008
- [D1] L. Drewnowski, Boundedness of vector measures with values in the spaces ${L}_{0}$ of Bochner measurable functions, Proc. Amer. Math. Soc. 91 (1984), 581-588. Zbl0601.28007
- [D2] L. Drewnowski, Topological vector groups and the Nevanlinna class, Funct. Approx. 22 (1994), 25-39.
- [DFP] L. Drewnowski, M. Florencio and P. J. Paúl, Some new classes of rings of sets with the Nikodym property, in: Functional Analysis (Trier, 1994), de Gruyter, Berlin, 1996, 143-152.
- [DL1] L. Drewnowski and I. Labuda, Lacunary convergence of series in ${L}_{0}$, Proc. Amer. Math. Soc. 126 (1998), 1655-1659. Zbl0894.46020
- [DL2] L. Drewnowski and I. Labuda, The Orlicz-Pettis theorem for topological Riesz spaces, ibid., 823-825. Zbl0885.40002
- [DL3] L. Drewnowski and I. Labuda, Copies of ${c}_{0}$ and < ℓ∞> in topological Riesz spaces, Trans. Amer. Math. Soc. 350 (1998), 3555-3570. Zbl0903.46010
- [DL4] L. Drewnowski and I. Labuda, Topological vector spaces of Bochner measurable functions, submitted, 1999.
- [DL5] L. Drewnowski and I. Labuda, Subseries convergence of series in sequence spaces determined by some ideals in P(ℕ), in preparation.
- [EK] R. Estrada and R. P. Kanwal, Series that converge on sets of null density, Proc. Amer. Math. Soc. 97 (1986), 682-686. Zbl0592.40001
- [HJ] J. Hoffmann-Jørgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159-186. Zbl0265.60005
- [J] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981. Zbl0466.46001
- [Kw] S. Kwapień, On Banach spaces containing ${c}_{0}$, Studia Math. 52 (1974), 187-188. Zbl0295.60003
- [L1] I. Labuda, Denumerability conditions and Orlicz-Pettis type theorems, Comment. Math. 18 (1974), 45-49. Zbl0297.28013
- [L2] I. Labuda, Submeasures and locally solid topologies on Riesz spaces, Math. Z. 195 (1987), 179-196. Zbl0601.46006
- [L3] I. Labuda, Spaces of measurable functions, Comment. Math., Tomus spec. in honorem Ladislai Orlicz II 1979, 217-249.
- [MO] W. Matuszewska and W. Orlicz, A note on modular spaces. IX, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 801-808. Zbl0164.43002
- [Mu] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin, 1983.
- [NS] D. Noll and W. Stadler, Abstract sliding hump technique and characterization of barrelled spaces, Studia Math. 94 (1989), 103-120. Zbl0711.46004
- [O] W. Orlicz, On perfectly convergent series in certain function spaces, Prace Mat. 1 (1955), 393-414 (in Polish); English transl. in: W. Orlicz, Collected Papers, Part I, Polish Sci. Publ., Warszawa, 1988, 830-850. Zbl0066.35601
- [P] M. Paštéka, Convergence of series and submeasures on the set of positive integers, Math. Slovaca 40 (1990), 273-278. Zbl0755.40003
- [R] S. Rolewicz, Metric Linear Spaces, Polish Sci. Publ. & Reidel, Warszawa & Dordrecht, 1984. Zbl0226.46001
- [SF] J. J. Sember and A. R. Freedman, On summing sequences of 0's and 1's, Rocky Mountain J. Math. 11 (1981), 419-425. Zbl0496.40008
- [T] P. Turpin, Convexités dans les espaces vectoriels topologiques généraux, Dissertationes Math. 131 (1976).
- [W] W. Wnuk, Representations of Orlicz lattices, ibid. 235 (1984). Zbl0566.46018

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.