### Abstract

In this paper we investigate the relation between weak convergence of a sequence {μ_{n}} of probability measures on a Polish space S converging weakly to the probability measure μ and continuous, norm-bounded functions into a Banach space X. We show that, given a norm-bounded continuous function f: S → X, it follows that lim_{n→∞} ∫_{S}f dμ_{n} = ∫_{S} f dμ-the limit one has for bounded and continuous real (or complex)-valued functions on S. This result is then applied to the stability theory of Feynman's operational calculus where it is shown that the theory can be significantly improved over previous results.

Original language | English |
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Pages (from-to) | 279-294 |

Number of pages | 16 |

Journal | Mathematical Physics Analysis and Geometry |

Volume | 14 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2011 |

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### All Science Journal Classification (ASJC) codes

- Mathematical Physics
- Geometry and Topology

### Cite this

**Weak Convergence and Banach Space-Valued Functions : Improving the Stability Theory of Feynman's Operational Calculi.** / Nielsen, Lance.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Weak Convergence and Banach Space-Valued Functions

T2 - Improving the Stability Theory of Feynman's Operational Calculi

AU - Nielsen, Lance

PY - 2011/12

Y1 - 2011/12

N2 - In this paper we investigate the relation between weak convergence of a sequence {μn} of probability measures on a Polish space S converging weakly to the probability measure μ and continuous, norm-bounded functions into a Banach space X. We show that, given a norm-bounded continuous function f: S → X, it follows that limn→∞ ∫Sf dμn = ∫S f dμ-the limit one has for bounded and continuous real (or complex)-valued functions on S. This result is then applied to the stability theory of Feynman's operational calculus where it is shown that the theory can be significantly improved over previous results.

AB - In this paper we investigate the relation between weak convergence of a sequence {μn} of probability measures on a Polish space S converging weakly to the probability measure μ and continuous, norm-bounded functions into a Banach space X. We show that, given a norm-bounded continuous function f: S → X, it follows that limn→∞ ∫Sf dμn = ∫S f dμ-the limit one has for bounded and continuous real (or complex)-valued functions on S. This result is then applied to the stability theory of Feynman's operational calculus where it is shown that the theory can be significantly improved over previous results.

UR - http://www.scopus.com/inward/record.url?scp=82955187685&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=82955187685&partnerID=8YFLogxK

U2 - 10.1007/s11040-011-9097-z

DO - 10.1007/s11040-011-9097-z

M3 - Article

VL - 14

SP - 279

EP - 294

JO - Mathematical Physics Analysis and Geometry

JF - Mathematical Physics Analysis and Geometry

SN - 1385-0172

IS - 4

ER -