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A Theory of Net Convergence with Applications to Vector Lattices

  • Author / Creator
    O'Brien, Michael John
  • The theory of convergence structures delivers a promising foundation on which to study general notions of convergence. However, that theory has one striking feature that stands out against all others: it is described using the language of filters. This is contrary to how convergence is used in functional analysis, where one often prefers to work with nets, and this thesis reconciles the issue by introducing an equivalent theory of net convergence structures. Our approach has several advantages over other efforts to develop a convergence theory using nets. Most notably, we are able to translate between the languages of filter and net convergence structures. These results make the classical theory of filter convergence spaces more accessible for the working mathematician and provide a new angle for studying aspects of vector lattice theory. We demonstrate the value of this approach using order convergence in vector lattices. This leads to the novel concept of order compactness, and it is shown that order compact sets satisfy an analogue of the Heine-Borel theorem in atomic order complete vector lattices.

  • Subjects / Keywords
  • Graduation date
    Fall 2021
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/r3-sg78-0691
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.