Main Article Content
A vector lattice version of Radström's embedding theorem
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.
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.
