### Abstract

The space of real Borel measures (Formula presented.) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone (Formula presented.) 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 (Formula presented.) are compact and semiflows on (Formula presented.) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because (Formula presented.) is rarely complete and (Formula presented.) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on (Formula presented.) and continuous semiflows. Both topics prepare for a dynamical systems theory on (Formula presented.).

Original language | English (US) |
---|---|

Pages (from-to) | 1-34 |

Number of pages | 34 |

Journal | Positivity |

DOIs | |

State | Accepted/In press - May 25 2017 |

### Fingerprint

### Keywords

- Dynamical systems
- Homogeneous maps
- Krein–Rutman theorem
- Measures
- Ordered normed vector space
- Spectral radius

### ASJC Scopus subject areas

- Analysis
- Theoretical Computer Science
- Mathematics(all)

### Cite this

*Positivity*, 1-34. https://doi.org/10.1007/s11117-017-0503-z

**Measures under the flat norm as ordered normed vector space.** / Gwiazda, Piotr; Marciniak-Czochra, Anna; Thieme, Horst.

Research output: Contribution to journal › Article

*Positivity*, pp. 1-34. https://doi.org/10.1007/s11117-017-0503-z

}

TY - JOUR

T1 - Measures under the flat norm as ordered normed vector space

AU - Gwiazda, Piotr

AU - Marciniak-Czochra, Anna

AU - Thieme, Horst

PY - 2017/5/25

Y1 - 2017/5/25

N2 - The space of real Borel measures (Formula presented.) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone (Formula presented.) 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 (Formula presented.) are compact and semiflows on (Formula presented.) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because (Formula presented.) is rarely complete and (Formula presented.) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on (Formula presented.) and continuous semiflows. Both topics prepare for a dynamical systems theory on (Formula presented.).

AB - The space of real Borel measures (Formula presented.) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone (Formula presented.) 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 (Formula presented.) are compact and semiflows on (Formula presented.) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because (Formula presented.) is rarely complete and (Formula presented.) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on (Formula presented.) and continuous semiflows. Both topics prepare for a dynamical systems theory on (Formula presented.).

KW - Dynamical systems

KW - Homogeneous maps

KW - Krein–Rutman theorem

KW - Measures

KW - Ordered normed vector space

KW - Spectral radius

UR - http://www.scopus.com/inward/record.url?scp=85019665070&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019665070&partnerID=8YFLogxK

U2 - 10.1007/s11117-017-0503-z

DO - 10.1007/s11117-017-0503-z

M3 - Article

AN - SCOPUS:85019665070

SP - 1

EP - 34

JO - Positivity

JF - Positivity

SN - 1385-1292

ER -