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Fon-Che Liu

Print publication date: 2016

Print ISBN-13: 9780198790426

Published to Oxford Scholarship Online: January 2017

DOI: 10.1093/acprof:oso/9780198790426.001.0001

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Basic Principles of Linear Analysis

Basic Principles of Linear Analysis

Chapter:
(p.179) 5 Basic Principles of Linear Analysis
Source:
Real Analysis
Author(s):

Fon-Che Liu

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198790426.003.0005

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

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