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2 results

  • 7 - Completions and the Lebesgue Spaces Lp(Ω)

  • from Part II - Normed Linear Spaces
  • James C. Robinson, University of Warwick
  • Book:
    An Introduction to Functional Analysis
    Published online:
    21 March 2020
    Print publication:
    12 March 2020, pp 89-98

  • Summary

    We show that the space C^0 is not complete in the L^1 norm and use this to motivate the abstract completion of a normed space and the Lebesgue integral. We use this approach to define the family L^p of Lebesgue spaces as the completion of the space of continuous functions in the L^p norm, and we prove some properties of these spaces.

  • 1 - Topological Rings and Modules and Their Completions

  • Radoslav Dimitric
  • Book:
    Slenderness
    Published online:
    30 November 2018
    Print publication:
    06 December 2018, pp 41-90

  • Summary

    Linear topologies are of utmost importanceComplete objects and metrizable objects play a crucial role in characterizing slenderness.Different kinds of inversesystems, such as surjective, flabby, etc. are introduced here. Product topologies play a central role, as do Hausdorff and non-discrete topologies. Conditions ensuring that infinite sums are contained in an object are investigated.