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.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. Preliminaries 1(30) Linear Algebra 2(9) Metric Spaces 11(9) Lebesgue Integration 20(11) Normed Spaces 31(20) Examples of Normed Spaces 31(8) Finite-dimensional Normed Spaces 39(6) Banach Spaces 45(6) Inner Product Spaces, Hilbert Spaces 51(36) Inner Products 51(9) Orthogonality 60(5) Orthogonal Complements 65(7) Orthonormal Bases in Infinite Dimensions 72(10) Fourier Series 82(5) Linear Operators 87(34) Continuous Linear Transformations 87(9) The Norm of a Bounded Linear Operator 96(8) The Space B(X, Y) 104(4) Inverses of Operators 108(13) Duality and the Hahn--Banach Theorem 121(46) Dual Spaces 121(6) Sublinear Functional, Seminorms and the Hahn--Banach Theorem 127(5) The Hahn--Banach Theorem in Normed Spaces 132(5) The General Hahn--Banach theorem 137(7) The Second Dual, Reflexive Spaces and Dual Operators 144(11) Projections and Complementary Subspaces 155(4) Weak and Weak-* Convergence 159(8) Linear Operators on Hilbert Spaces 167(38) The Adjoint of an Operator 167(9) Normal, Self-adjoint and Unitary Operators 176(7) The Spectrum of an Operator 183(9) Positive Operators and Projections 192(13) Compact Operators 205(30) Compact Operators 205(11) Spectral Theory of Compact Operators 216(10) Self-adjoint Compact Operators 226(9) Integral and Differential Equations 235(30) Fredholm Integral Equations 235(10) Volterra Integral Equations 245(2) Differential Equations 247(6) Eigenvalue Problems and Green's Functions 253(12) Solutions to Exercises 265(50) Further Reading 315(2) References 317(2) Notation Index 319(2) Index 321