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Some Remarks on the Algebraic Sum of Ideals and Riesz Subspaces

  Published:2011-08-03
 Printed: Jun 2013
  • Witold Wnuk,
    Faculty of Mathematics and Computer Science, A. Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland
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Abstract

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, $I \cap G = \{0\}$, and $I + G$ is not closed.
Keywords: locally solid Riesz space, Riesz subspace, ideal, minimal topological vector space, Lebesgue property locally solid Riesz space, Riesz subspace, ideal, minimal topological vector space, Lebesgue property
MSC Classifications: 46A40, 46B42, 46B45 show english descriptions Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42]
Banach lattices [See also 46A40, 46B40]
Banach sequence spaces [See also 46A45]
46A40 - Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42]
46B42 - Banach lattices [See also 46A40, 46B40]
46B45 - Banach sequence spaces [See also 46A45]
 

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