Measures under the flat norm as ordered normed vector space

• Verfasst von: Gwiazda, Piotr [VerfasserIn]
• Verfasst von: Marciniak-Czochra, Anna [VerfasserIn]
• Verfasst von: Thieme, Horst R. [VerfasserIn]
• Titel: Measures under the flat norm as ordered normed vector space
• Verf.angabe: Piotr Gwiazda, Anna Marciniak-Czochra, Horst R. Thieme
• Jahr des Originals: 2017
• Umfang: 34 S.
• Fussnoten: Gesehen am 28.06.2018 ; Published online: 25 May 2017
• Titel Quelle: Enthalten in: Positivity
• Jahr Quelle: 2018
• Band/Heft Quelle: 22(2018), 1, S. 105-138
• ISSN Quelle: 1572-9281
• DOI: doi:10.1007/s11117-017-0503-z
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1561837393

Abstract

The space of real Borel measures M(S)M(S)\mathcal {M}(S) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone M+(S)M+(S)\mathcal {M}_+(S) of nonnegative measures, is considered from an ordered normed vector space perspective in order to apply the well-developed theory of this area. The flat norm is considered in place of the variation norm because subsets of M+(S)M+(S)\mathcal {M}_+(S) are compact and semiflows on M+(S)M+(S)\mathcal {M}_+(S) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because M(S)M(S)\mathcal {M}(S) is rarely complete and M+(S)M+(S)\mathcal {M}_+(S) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on M+(S)M+(S)\mathcal {M}_+(S) and continuous semiflows. Both topics prepare for a dynamical systems theory on M+(S)M+(S)\mathcal {M}_+(S).