Functiones et Approximatio Commentarii Mathematici

On infinite sums of closed ideals in $F}-lattices

Lech Drewnowski

Full-text: Open access

Abstract

The main result of the paper is that if $(I_n)$ is a sequence of closed ideals in an $F$-lattice $E$, then also $\sum_{n=1}^\infty I_n$, the set of all elements $x\in E$ of the form $x=\sum_n x_n$, where $x_n\in I_n$ for every $n$, is a closed ideal in $E$.

Article information

Source
Funct. Approx. Comment. Math., Volume 44, Number 2 (2011), 279-284.

Dates
First available in Project Euclid: 22 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.facm/1308749131

Digital Object Identifier
doi:10.7169/facm/1308749131

Mathematical Reviews number (MathSciNet)
MR2841186

Zentralblatt MATH identifier
1229.46005

Subjects
Primary: 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) 46A40: Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42] 46B42: Banach lattices [See also 46A40, 46B40]

Keywords
$F$-latices closed ideals infinite sums of ideals

Citation

Drewnowski, Lech. On infinite sums of closed ideals in $F}-lattices. Funct. Approx. Comment. Math. 44 (2011), no. 2, 279--284. doi:10.7169/facm/1308749131. https://projecteuclid.org/euclid.facm/1308749131


Export citation

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., 2003.
  • N. Bourbaki, Topologie Générale, Ch. III–VIII, Hermann, Paris, 1960.
  • P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, Berlin, 1991.
  • H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, Berlin, 1974.
  • W. Wnuk, Some remarks on the algebraic sum of ideals and Riesz subspaces, Canadian Math. Bull. (to appear).
  • W. Wnuk and B. Wiatrowski, Order properties of quotient Riesz spaces, Atti Sem. Mat. Fis. Univ. Modena e Reggio Emilia 53 (2005), 417–428.