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An isomorphic characterization of L 1-spaces.
- Source :
- Indagationes Mathematicae; Dec2007, Vol. 18 Issue 4, p629-640, 12p
- Publication Year :
- 2007
-
Abstract
- Abstract: We show that a sequentially (τ)-complete topological vector lattice X <subscript>τ</subscript> is isomorphic to some L <superscript>1</superscript>(μ), if and only if the positive cone can be written as X <subscript>+</subscript> = ℝ<subscript>+</subscript> B for some convex, (τ)-bounded, and (τ)-closed set B ⊂ X <subscript>+</subscript> \ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y <subscript>+</subscript> = ℝ<subscript>+</subscript> (for ‖u‖ >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials. [Copyright &y& Elsevier]
- Subjects :
- VECTOR spaces
RIESZ spaces
ISOMORPHISM (Mathematics)
BANACH lattices
Subjects
Details
- Language :
- English
- ISSN :
- 00193577
- Volume :
- 18
- Issue :
- 4
- Database :
- Supplemental Index
- Journal :
- Indagationes Mathematicae
- Publication Type :
- Academic Journal
- Accession number :
- 33661176
- Full Text :
- https://doi.org/10.1016/S0019-3577(07)80068-1