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Measures under the flat norm as ordered normed vector space

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...

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Hauptverfasser: Gwiazda, Piotr (VerfasserIn) , Marciniak-Czochra, Anna (VerfasserIn) , Thieme, Horst R. (VerfasserIn)
Dokumenttyp: Article (Journal)
Sprache:Englisch
Veröffentlicht: 2018
In: Positivity
Year: 2017, Jahrgang: 22, Heft: 1, Pages: 105-138
ISSN:1572-9281
DOI:10.1007/s11117-017-0503-z
Online-Zugang:Verlag, Volltext: http://dx.doi.org/10.1007/s11117-017-0503-z
Verlag, Volltext: https://link.springer.com/article/10.1007/s11117-017-0503-z
Verlag, Volltext: https://link.springer.com/content/pdf/10.1007%2Fs11117-017-0503-z.pdf
Volltext
Verfasserangaben:Piotr Gwiazda, Anna Marciniak-Czochra, Horst R. Thieme

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520 |a 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). 
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