# Linear Functional Analysis

Springer Science & Business Media, 2000 - Mathematics - 273 pages
This introduction to the ideas and methods of linear functional analysis shows how familiar and useful concepts from finite-dimensional linear algebra can be extended or generalized to infinite-dimensional spaces. Aimed at advanced undergraduates in mathematics and physics, the book assumes a standard background of linear algebra, real analysis (including the theory of metric spaces), and Lebesgue integration, although an introductory chapter summarizes the requisite material.

The initial chapters develop the theory of infinite-dimensional normed spaces, in particular Hilbert spaces, after which the emphasis shifts to studying operators between such spaces. Functional analysis has applications to a vast range of areas of mathematics; the final chapters discuss the particularly important areas of integral and differential equations.

Further highlights of the second edition include:

a new chapter on the Hahn???Banach theorem and its applications to the theory of duality. This chapter also introduces the basic properties of projection operators on Banach spaces, and weak convergence of sequences in Banach spaces - topics that have applications to both linear and nonlinear functional analysis;

extended coverage of the uniform boundedness theorem;

plenty of exercises, with solutions provided at the back of the book.

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### Contents

 Preliminaries 1 11 Linear Algebra 2 12 Metric Spaces 11 13 Lebesgue Integration 20 Normed Spaces 31 22 Finitedimensional Normed Spaces 39 23 Banach Spaces 45 Inner Product Spaces Hilbert Spaces 51
 52 Normal Selfadjoint and Unitary Operators 132 53 The Spectrum of an Operator 139 54 Positive Operators and Projections 148 Compact Operators 161 62 Spectral Theory of Compact Operators 172 63 Selfadjoint Compact Operators 182 Integral and Differential Equations 191 72 Volterra Integral Equations 201

 32 Orthogonality 60 33 Orthogonal Complements 65 34 Orthonormal Bases in Infinite Dimensions 72 35 Fourier Series 82 Linear Operators 87 42 The Norm of a Bounded Linear Operator 96 43 The Space BX Y and Dual Spaces 104 44 Inverses of Operators 111 Linear Operators on Hilbert Spaces 123
 73 Differential Equations 203 74 Eigenvalue Problems and Greens Functions 208 Solutions to Exercises 221 Further Reading 265 References 267 Notation Index 269 Index 271 Copyright