Introduction to Functional Analysis

Analyzes the theory of normed linear spaces and of linear mappings between such spaces, providing the necessary foundation for further study in many areas of analysis. Strives to generate an appreciation for the unifying power of the abstract linear-space point of view in surveying the problems of linear algebra, classical analysis, and differential and integral equations. This second edition incorporates recent developments in functional analysis to make the selection of topics more appropriate for current courses in functional analysis. Additions to this new edition include: a chapter on Banach algebras, and material on weak topologies and duality, equicontinuity, the Krein-Milman theorem, and the theory of Fredholm operators. Greater emphasis is also placed on closed unbounded linear operators, with more illustrations drawn from ordinary differential equations. Title Page......Page 3 Copyright Information......Page 4 Dedication......Page 5 Preface......Page 6 Contents......Page 9 Introduction......Page 13 I The Abstract Approach to Linear Problems......Page 16 1.1 Abstract Linear Spaces......Page 17 1.2 Examples of Linear Spaces......Page 22 1.3 Linear Operators......Page 25 1.4 Linear Operators in Finite-Dimensional Spaces......Page 30 1.5 Other Examples of Linear Operators......Page 33 1.6 Direct Sums and Quotient Spaces......Page 40 1.7 Linear Functionals......Page 43 1.8 Linear Functionals in Finite-Dimensional Spaces......Page 47 1.9 Zorn's Lemma......Page 49 1.10 Extension Theorems for Linear Operators......Page 50 1.11 Hamel Bases......Page 53 1.12 The Transpose of a Linear Operator......Page 56 1.13 Annihilators, Ranges, and Null Spaces......Page 57 1.14 Conclusions......Page 61 II Topological Linear Spaces......Page 63 II.1 Normed Linear Spaces......Page 64 II.2 Examples of Normed Linear Spaces......Page 68 II.3 Finite-Dimensional Normed Linear Spaces......Page 74 II.4 Banach Spaces......Page 78 II.5 Quotient Spaces......Page 83 II.6 Inner-Product Spaces......Page 85 II.7 Hilbert Space......Page 98 II.8 Examples of Complete Orthonormal Sets......Page 103 II.9 Topological Linear Spaces......Page 106 II.10 Convex Sets......Page 112 II.11 Locally Convex Spaces......Page 117 II.12 Minkowski Functionals......Page 123 II.13 Metrizable Topological Linear Spaces......Page 127 III Linear Functionals and Weak Topologies......Page 133 III.1 Linear Varieties and Hyperplanes......Page 134 III.2 The Hahn-Banach Theorem......Page 137 III.3 The Conjugate of a Normed Linear Space......Page 146 III.4 The Second Conjugate Space......Page 151 III.5 Some Representations of Linear Functionals......Page 153 III.6 Weak Topologies for Linear Spaces......Page 168 III.7 Polar Sets and Annihilators......Page 172 III.8 Equicontinuity and S-topologies......Page 177 III.9 The Principle of Uniform Boundedness......Page 181 III.10 Weak Topologies for Normed Linear Spaces......Page 184 III.11 The Krein-Milman Theorem......Page 193 IV General Theorems on Linear Operators......Page 200 IV. 1 Spaces of Linear Operators......Page 201 IV.2 Integral Equations of the Second Kind......Page 208 IV.3 L^2 Kernels......Page 213 IV.4 Differential Equations and Integral Equations......Page 217 IV.5 Closed Linear Operators......Page 220 IV.6 Some Representations of Bounded Linear Operators......Page 231 IV.7 The M. Riesz Convexity Theorem......Page 236 IV.8 Coniugates of Linear Operators......Page 238 IV.9 Theorems About Continuous Inverses......Page 246 IV.10 The States of an Operator and Its Conjugate......Page 249 IV.11 Ad]oint Operators......Page 254 IV.12 Projections......Page 258 IV.13 Fredholm Operators......Page 265 V Spectral Analysis of Linear Operators......Page 276 V.1 Analytic Vector-Valued Functions......Page 277 V.2 The Resolvent Operator......Page 284 V.3 The Spectrum of a Bounded Linear Operator......Page 289 V.4 Subdivisions of the Spectrum......Page 294 V.5 Reducibility......Page 299 V.6 The Ascent and Descent of an Operator......Page 301 V.7 Compact Operators......Page 305 V.8 An Operational Calculus......Page 321 V.9 Spectral Sets. The Spectral Mapping Theorem......Page 332 V.10 Isolated Points of the Spectrum......Page 340 V.11 Operators with a Rational Resolvent......Page 348 Vl Spectral Analysis in Hilbert Space......Page 353 VI.1 Bilinear and Quadratic Forms......Page 354 VI.2 Symmetric Operators......Page 357 VI.3 Normal and Self-adjoint Operators......Page 361 VI.4 Compact Symmetric Operators......Page 365 VI.5 Symmetric Operators with Compact Resolvent......Page 373 VI.6 The Spectral Theorem for Bounded Self-adjoint Operators......Page 375 VI.7 Unitary Operators......Page 386 VI.8 Unbounded Self-adjoint Operators......Page 392 VII Banach Algebras......Page 398 VII.1 Examples of Banach Algebras......Page 399 VII.2 Spectral Theory in a Banach Algebra......Page 405 VII.3 Ideals and Homomorphisms......Page 412 VII.4 Commutative Banach Algebras......Page 416 VII.5 Applications and Extensions of the Gelland Theory......Page 427 VII.6 B*-algebras......Page 438 VII.7 The Spectral Theorem for a Normal Operator......Page 442 Bibliography......Page 457 List of Special Symbols......Page 467 Index......Page 471