A vector lattice version of Radström's embedding theorem

  • CCA Labuschagne School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
  • AL Pinchuck School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
  • CJ van Alten School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
Keywords: Banach space, Banach lattice, Hausdorff metric, hyperspace, Radstr&#246m's embedding, Riesz space, vector lattice

Abstract

Radström's embedding theorem for ‘near vector spaces', which are essentially vector spaces without additive inverses, is extended to embeddings of ‘near vector lattices', which are essentially vector lattices without additive inverses, into vector lattices.
If the ‘near vector space' is endowed with a metric, properties on the metric are considered for which the norm completion of the embedding space is one of the classical Banach spaces C Ω) or Lp(μ) for 1 ≤ p < ∞. This order embedding procedure is then applied to the hyperspaces:

• cbf(X) of nonempty convex bounded closed subsets of X,
• cwk(X) of nonempty convex weakly compact subsets of X,
• ck(X) of nonempty convex compact subsets of X,
where X, is a Banach space.

Published
2007-10-10
Section
Articles

Journal Identifiers


eISSN: 1727-933X
print ISSN: 1607-3606