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5 results

  • 6 - Linear Operators

  • from Part II - Elements Of Classical Functional Analysis
  • Martin Buntinas, Loyola University, Chicago
  • Book:
    Classical and Discrete Functional Analysis with Measure Theory
    Published online:
    06 January 2022
    Print publication:
    20 January 2022, pp 214-252

  • Summary

    Linear Operators and Linear Functionals are studied. Then Operator Spaces, Topological Duals, and Second Duals of Normed Spaces are considered. Lebesgue L(p) Spaces are defined and studied. The theorems of Hahn–Banach Extension, Baire Category, Riesz Representation, Open Mapping, Closed Graph, and Banach Fixed Points are all proven.

Counterexamples in Measure and Integration
  • René L. Schilling, Franziska Kühn
  • Published online:
    27 May 2021
    Print publication:
    17 June 2021
  • Often it is more instructive to know 'what can go wrong' and to understand 'why a result fails' than to plod through yet another piece of theory. In this text, the authors gather more than 300 counterexamples - some of them both surprising and amusing - showing the limitations, hidden traps and pitfalls of measure and integration. Many examples are put into context, explaining relevant parts of the theory, and pointing out further reading. The text starts with a self-contained, non-technical overview on the fundamentals of measure and integration. A companion to the successful undergraduate textbook Measures, Integrals and Martingales, it is accessible to advanced undergraduate students, requiring only modest prerequisites. More specialized concepts are summarized at the beginning of each chapter, allowing for self-study as well as supplementary reading for any course covering measures and integrals. For researchers, it provides ample examples and warnings as to the limitations of general measure theory. This book forms a sister volume to René Schilling's other book Measures, Integrals and Martingales (www.cambridge.org/9781316620243).

  • 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.

  • STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES

  • Part of
  • SAEID MAGHSOUDI, RASOUL NASR-ISFAHANI
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 84 / Issue 3 / December 2011
    Published online by Cambridge University Press:
    06 September 2011, pp. 504-515
    Print publication:
    December 2011
    • Article
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  • Let X be a locally compact space, and 𝔏∞0(X,ι) be the space of all essentially bounded ι-measurable functions f on X vanishing at infinity. We introduce and study a locally convex topology β1(X,ι) on the Lebesgue space 𝔏1(X,ι) such that the strong dual of (𝔏1(X,ι),β1(X,ι)) can be identified with . Next, by showing that β1(X,ι) can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove that L1 (G) , the group algebra of a locally compact Hausdorff topological group G, equipped with the convolution multiplication is a complete semitopological algebra under the β1 (G) topology.