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Theoretical Foundations

  • James LottesEmail author
Chapter
  • 466 Downloads
Part of the Springer Theses book series (Springer Theses)

Abstract

This chapter is background, introducing the abstract framework used by later chapters. Covered topics include sesquilinear forms on (reflexive) Banach spaces, sectorial forms, their Cartesian decompositions, and the Lax-Milgram lemma. The Lax-Milgram lemma, which provides sufficient conditions for an operator associated with a sesquilinear form to have a bounded inverse, will be used heavily in later chapters. In particular, its premises will serve to define the scope of problems that the analysis of later chapters applies to, namely to operators with positive real part. The version of the lemma presented is a generalization due to Roşca to reflexive Banach spaces. Otherwise the material is mostly standard, but serves to set the notation used throughout the book.

Keywords

Hilbert Space Quadratic Form Duality Pairing Multigrid Method Positive Real Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society (1998)Google Scholar
  2. 2.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Heidelberg (1966)Google Scholar
  3. 3.
    Roşca, I.: Bilinear coercive and weakly coercive operators. An. Univ. Bucureşti Mat. (2), 183–188 (2002)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Google Inc.Mountain ViewUSA

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