Illinois Journal of Mathematics

Solid sequence $F$-spaces of $L_0$-type over submeasures on $\mathbb{N}$

Lech Drewnowski and Iwo Labuda

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Abstract

We study solid sequence $F$-spaces $\lambda_{0}(\eta)$, nonseparable in general, and their closed separable subspaces $\lambda_{00}(\eta)$. The space $\lambda_{0}(\eta)$ is associated with a strictly positive submeasure $\eta$ on $\mathbb{N}$ and equipped with the topology of convergence in submeasure. While $\lambda_{0}(\eta)$'s may be viewed as analogs of usual $L_0$-spaces, the relation between $\lambda_{00}(\eta)$ and $\lambda_{0}(\eta )$ often resembles that between $c_0$ and $l_\infty$. For many $\eta$'s, the weak topology of these spaces coincides with that of coordinate-wise convergence, they are not locally pseudoconvex and yet have the Bounded Multiplier Property. Further, in agreement with the analogy to $L_0=L_0[0,1]$, they possess copies of $l_p$ for $0 \lt p \leq 2$, and yet in contrast to $L_0$ they contain a lot of well-located copies of $c_0$ and $l_\infty$; also, the quotient $\lambda_{0}(\eta)/\lambda_{00}(\eta)$ contains a copy of $L_0$. All of this happens already for the spaces $\lambda_{0}=\lambda _{0}(\bar d)$ and $\lambda_{00}=\lambda_{00}(\bar d)$ with $\bar d$ being a submeasure closely related to the standard density $d$, in which case, moreover: (1) There is a series in $\lambda_{00}$ all of whose subseries of density zero are convergent, and yet its partial sums are unbounded. (2) The Orlicz–Pettis theorem fails in $\lambda_{0}$. (3) $\lambda_{00}$ can be used to show that some earlier constructed normed barrelled spaces are not ultrabarrelled.

Article information

Source
Illinois J. Math., Volume 53, Number 2 (2009), 623-678.

Dates
First available in Project Euclid: 23 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1266934797

Digital Object Identifier
doi:10.1215/ijm/1266934797

Mathematical Reviews number (MathSciNet)
MR2594648

Zentralblatt MATH identifier
1200.46008

Subjects
Primary: 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) 46A45: Sequence spaces (including Köthe sequence spaces) [See also 46B45] 28A12: Contents, measures, outer measures, capacities

Citation

Drewnowski, Lech; Labuda, Iwo. Solid sequence $F$-spaces of $L_0$-type over submeasures on $\mathbb{N}$. Illinois J. Math. 53 (2009), no. 2, 623--678. doi:10.1215/ijm/1266934797. https://projecteuclid.org/euclid.ijm/1266934797


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References

  • C. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces with applications to economics, 2nd ed., Math. Surveys and Monographs, vol. 105, Amer. Math. Soc., Providence, RI, 2003.
  • C. Bessaga, A. Pełczyński and S. Rolewicz, Some properties of the space $(s)$, Colloq. Math. 7 (1957), 45–51.
  • L. Drewnowski, Topological rings of sets, continuous set functions, integration. I, II, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys. 20 (1972), 269–286.
  • L. Drewnowski, Un théorème sur les opérateurs de $l_{\infty}(\Gamma)$, C. R. Acad. Sci. Paris Sér. A 281 (1975), 967–969.
  • L. Drewnowski, An extension of a theorem of Rosenthal on operators acting from $l_{\infty}(\Gamma)$, Studia Math. 57 (1976), 209–215.
  • L. Drewnowski, On minimally subspace-comparable $F$-spaces, J. Funct. Anal. 26 (1977), 315–332.
  • L. Drewnowski, $F$-spaces with a basis which is shrinking but not hyper-shrinking, Studia Math. 64 (1979), 97–104.
  • L. Drewnowski, On minimal topological linear spaces and strictly singular operators, Comment. Math. Prace Mat. Special Issue II (1979), 89–106.
  • L. Drewnowski, Topological vector groups and the Nevanlinna class, Funct. Approx. 22 (1994), 25–39.
  • L. Drewnowski, M. Florencio and P. J. Paúl, Barrelled subspaces of spaces with subseries decompositions or Boolean rings of projections, Glasg. Math. J. 36 (1994), 57–69.
  • L. Drewnowski, M. Florencio and P. J. Paúl, Some new classes of rings of sets with the Nikodym property, Proceedings of the 1st international workshop on functional analysis at Trier University, Sept. 26–Oct. 1, 1994, de Gruyter, Berlin–New York, 1996, pp. 143–152.
  • L. Drewnowski and I. Labuda, Copies of $c_0$ and $l_\infty$ in topological Riesz spaces, Trans. Amer. Math. Soc. 350 (1998), 3555–3570.
  • L. Drewnowski and I. Labuda, Vector series whose lacunary subseries converge, Studia Math. 138 (2000), 53–80.
  • L. Drewnowski and T. Łuczak, On nonatomic submeasures on $\mathbb{N}$, Arch. Math. 91 (2008), 76–85.
  • L. Drewnowski and T. Łuczak, On nonatomic submeasures on $\mathbb{N}$. II, J. Math. Anal. Appl. 347 (2008), 442–449.
  • L. Drewnowski and M. Nawrocki, Connectedness in some topological vector-lattice groups of sequences, Math. Scand. (2010) in print.
  • L. Drewnowski and P. J. Paúl, The Nikodym property for ideals of sets defined by matrix summability methods, Rev. R. Acad. Cienc. Exactas Fis. Nat. (Esp) 94 (2000), 485–503.
  • R. Filipow, N. Mrożek, I. Recław and P. Szuca, Ideal convergence of bounded sequences, J. Symbolic Logic 72 (2007), 501–512.
  • H. Jarchow, Locally convex spaces, Teubner, Stuttgart, 1981.
  • N. J. Kalton, Linear operators on $L_p$, $0<p<1$, Trans. Amer. Math. Soc. 259 (1980), 319–355.
  • N. J. Kalton, N. T. Peck and J. W. Roberts, An $F$-space sampler, London Math. Soc. Lecture Notes, vol. 89, Cambridge Univ. Press, Cambridge, 1984.
  • M. Kanter, Stable laws and the imbedding of $L^p$ spaces, Amer. Math. Monthly 80 (1973), 403–407.
  • S. Kwapień, On a theorem of L. Schwartz and its applicatioons to absolutely summing operators, Studia Math. 38 (1970), 193–201.
  • I. Labuda, On bounded multiplier convergence of series in Orlicz spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1974), 651–655.
  • M. Nawrocki, On the Orlicz–Pettis property in nonlocally convex $F$-spaces, Proc. Amer. Math. Soc. 101 (1987), 492–496.
  • M. Nawrocki The Orlicz–Pettis theorem fails for Lumer's Hardy spaces $(LH)^p(B)$, Proc. Amer. Math. Soc. 109 (1990), 957–963.
  • S. Rolewicz, Metric linear spaces, Polish Scientific Publishers & D. Reidel Publishing Co., Warszawa/Dordrecht, Boston, Lancaster, 1984.
  • S. Solecki, Analytic ideals, Bull. Symbolic Logic 2 (1996), 339–348.
  • S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic 99 (1999), 51–72.
  • Ph. Turpin, Convexités dans les espaces vectoriels topologiques généraux, Dissert. Math. 131 (1976) 1–221.
  • P. Wojtaszczyk, Banach spaces for analysts, Cambridge Univ. Press, Cambridge, 1991.