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Some Remarks on the Algebraic Sum of Ideals and Riesz Subspaces

Published online by Cambridge University Press:  20 November 2018

Witold Wnuk
Witold Wnuk*
Affiliation:
Faculty of Mathematics and Computer Science, A.Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland e-mail: [email protected]
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Abstract.

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Following ideas used by Drewnowski and Wilansky we prove that if $I$ is an infinite dimensional and infinite codimensional closed ideal in a complete metrizable locally solid Riesz space and $I$ does not contain any order copy of ${{\mathbb{R}}^{\mathbb{N}}}$ then there exists a closed, separable, discrete Riesz subspace $G$ such that the topology induced on $G$ is Lebesgue, $I\,\bigcap \,G\,=\,\left\{ 0 \right\}$, and $I\,+\,G$ is not closed.

Keywords

46A4046B4246B45locally solid Riesz spaceRiesz subspaceidealminimal topological vector spaceLebesgue property
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 2 , 01 June 2013 , pp. 434 - 441
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Aliprantis, C. and Burkinshaw, O., Locally solid Riesz spaces with applications to economics. Mathematical Surveys and Monographs, 105, American Mathematical Society, Providence, 2003.Google Scholar
[2] Drewnowski, L., On minimal topological linear spaces and strictly singular operators. Comment. Math. Special Issue 2(1979), 89–106. Google Scholar
[3] Drewnowski, L., Quasi-complements in F-spaces. Studia Math. 77(1984), 373–391. Google Scholar
[4] Kalton, N. J., The basic sequence problem. Studia Math. 116(1995), no. 2, 167–187. Google Scholar
[5] Luxemburg, W. A. J. and Zaanen, A. C., Riesz spaces I, North-Holland Mathematical Library, North-Holland, Amsterdam-London, 1971.Google Scholar
[6] Meyer-Nieberg, P., Banach lattices. Universitext, Springer-Verlag, Berlin, 1991.Google Scholar
[7] Peck, N. T., On non locally convex spaces. II. Math. Ann. 178(1968), 209–218. http://dx.doi.org/10.1007/BF01350661 Google Scholar
[8] Schaefer, H. H., Banach lattices and positive operators. Die Grundlehren der mathematischen Wissenschaften, 215, Springer-Verlag, New York-Heidelberg, 1974.Google Scholar
[9] Wiatrowski, B., On a structure of quotient Riesz spaces. (Polish) Ph.D. Thesis, A. Mickiewicz University, Poznan, 2006.Google Scholar
[10] Wilansky, A., Semi-Fredholm maps of FK-spaces. Math. Z. 144(1975), no. 1, 9–12. http://dx.doi.org/10.1007/BF01214402 Google Scholar
[11] Wnuk, W., The maximal topological extension of a locally solid Riesz space with the Fatou property. Comment. Math. Prace Mat. 24(1984), no. 1, 167–172. Google Scholar
[12] Wnuk, W., Selected topics in quotient Riesz spaces–a short survey. In: Banach and function spaces II, Yokohama Publ., Yokohama, 2008, pp. 243–276. Google Scholar
[13] Wnuk, W. and Wiatrowski, B., Order properties of quotient Riesz spaces. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 53(2005), no. 2, 417–42Google Scholar