Abstract
According to Corollary 20.3 the linear functional φ on the Riesz space E is order bounded if and only if φ is regular, i.e., if and only if φ = φl - φ2 with φ1 and φ2 positive. The vector space E~ of all order bounded linear functionals on E is a Dedekind complete Riesz space with respect to the ordering defined by saying that φl ≤ φ2 holds whenever φ2 - φ1 is positive. It follows that for any φ ∈ E~ the functionals φ+ = φ ∨ 0 and φ- = (-φ) ∨ 0 exist in E~ and φ = φ+ - φ -. Also, | φ |= φ ∨ (-φ) exists in E~ and | φ |= φ+ + φ-. The space E~ is called the order dual of E (see Corollary 20.3).
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© 1997 Springer-Verlag Berlin Heidelberg
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Zaanen, A.C. (1997). Order Duals and Adjoint Operators. In: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60637-3_13
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DOI: https://doi.org/10.1007/978-3-642-60637-3_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-64487-0
Online ISBN: 978-3-642-60637-3
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