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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.
KeywordsHilbert Space Quadratic Form Duality Pairing Multigrid Method Positive Real Part
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