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Bochner integrals in ordered vector spaces.

A C M van Rooij1, W B van Zuijlen2

  • 11Department of Mathematics, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands.

Positivity
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Summary
This summary is machine-generated.

Researchers developed a method to cover Archimedean directed ordered vector spaces with Banach spaces. This extends Bochner integrability, creating an order-preserving integral for functions in these spaces.

Keywords:
Bochner integralClosed coneGenerating coneOrdered Banach spaceOrdered vector space

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Area of Science:

  • Functional Analysis
  • Real Analysis
  • Vector Spaces

Background:

  • Archimedean directed ordered vector spaces are fundamental in mathematical analysis.
  • Bochner integrability is a key concept for functions with vector-valued outputs.
  • Existing methods for integrating functions in ordered vector spaces have limitations.

Purpose of the Study:

  • To introduce a novel method for covering Archimedean directed ordered vector spaces using Banach spaces.
  • To extend the definition and properties of Bochner integrability to functions mapping into these vector spaces.
  • To establish the properties of the resulting space of integrable functions and the integral operator.

Main Methods:

  • Developing a natural covering of an Archimedean directed ordered vector space E by Banach spaces.
  • Generalizing the concept of Bochner integrability for functions with codomain E.
  • Characterizing the space of integrable functions and the integral as an order-preserving map.

Main Results:

  • A natural covering of Archimedean directed ordered vector spaces by Banach spaces was established.
  • The notion of Bochner integrability was successfully extended to functions with values in E.
  • The set of integrable functions forms an Archimedean directed ordered vector space.
  • The integral was shown to be an order-preserving map.

Conclusions:

  • The proposed method provides a robust framework for integrating functions in Archimedean directed ordered vector spaces.
  • The extension of Bochner integrability offers new analytical tools for functions with vector-valued outputs.
  • The integral's order-preserving property is crucial for applications in various mathematical fields.