Chapter

Basic Principles of Linear Analysis

Fon-Che Liu

in Real Analysis

Published in print October 2016 | ISBN: 9780198790426
Published online January 2017 | e-ISBN: 9780191831676 | DOI: https://dx.doi.org/10.1093/acprof:oso/9780198790426.003.0005

Series: Oxford Graduate Texts in Mathematics

Basic Principles of Linear Analysis

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This chapter on the basic principles of linear analysis presents elements of functional analysis that are frequently used in setting up spaces on which certain operators intimately related to the problems at issue can be studied with a firm base. The most fundamental are the Baire category theorem and the separation principle. The Baire category theorem manifests itself in the uniform boundedness principle, open mapping theorem, and closed graph theorem; while the separation principle is applied in the name of Hahn–Banach. Our treatment of the separation principle is more geometrical than usual. In the latter part of the chapter, emphasis is put on geometric aspects by introducing Hilbert space in which a concept of orthogonal projection plays a leading role. The Riesz representation for bounded linear functionals and Fourier expansion with respect to an orthonormal basis are the main components of the theory addressed.

Keywords: linear analysis; Baire category; uniform boundedness; open mapping; closed graph; Riesz representation; orthonormal basis; Fourier expansion

Chapter.  30959 words. 

Subjects: Mathematics

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