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. Cover......Page 1 S Title......Page 2 INTRODUCTION TO FUNCTIONAL ANALYSIS, SECOND EDITION......Page 4 (QA320.T3 1986) 515.7......Page 5 Dedication......Page 6 PREFACE......Page 7 CONTENTS......Page 10 FUNCTIONS......Page 14 INEQUALITIES......Page 15 I THE ABSTRACT APPROACH TO LINEAR PROBLEMS......Page 17 1.1 ABSTRACT LINEAR SPACES......Page 18 1.2. EXAMPLES OF LINEAR SPACES......Page 23 1.3 LINEAR OPERATORS......Page 26 1.4 LINEAR OPERATORS IN FINITE-DIMENSIONAL SPACES......Page 31 1.5 OTHER EXAMPLES OF LINEAR OPERATORS......Page 34 1.6 DIRECT SUMS AND QUOTIENT SPACES......Page 41 1.7 LINEAR FUNCTIONALS......Page 44 1.8 LINEAR FUNCTIONALS IN FINITE-DIMENSIONAL SPACES......Page 48 1.9 ZORN'S LEMMA......Page 50 1.10 EXTENSION THEOREMS FOR LINEAR OPERATORS......Page 51 1.11 HAMEL BASES......Page 54 1.12 THE TRANSPOSE OF A LINEAR OPERATOR......Page 57 1.13 ANNIHILATORS, RANGES, AND NULL SPACES......Page 58 1.14 CONCLUSIONS......Page 62 II TOPOLOGICAL LINEAR SPACES......Page 64 2.1 NORMED LINEAR SPACES......Page 65 2.2 EXAMPLES OF NORMED LINEAR SPACES......Page 69 2.3 FINITE-DIMENSIONAL NORMED LINEAR SPACES......Page 75 2.4 BANACH SPACES......Page 79 2.5 QUOTIENT SPACES......Page 84 2.6 INNER-PRODUCT SPACES......Page 86 2.7 HuBERT SPACE......Page 99 2.8 EXAMPLES OF COMPLETE ORTHONORMAL SETS......Page 104 2.9 TOPOLOGICAL LINEAR SPACES......Page 107 2.10 CONVEX SETS......Page 113 2.11 LOCALLY CONVEX SPACES......Page 118 2.12 MINKOWSKI FUNCTIONALS......Page 124 2.13 METRIZABLE TOPOLOGICAL LINEAR SPACES......Page 128 III LINEAR FUNCTIONALS AND WEAK TOPOLOGIES......Page 134 3.1 LINEAR VARIETIES AND HYPERPLANES......Page 135 3.2 THE HAHN-BANACH THEOREM......Page 138 3.3 THE CONJUGATE OF A NORMED LINEAR SPACE......Page 147 3.4. THE SECOND CONJUGATE SPACE......Page 152 3.5. SOME REPRESENTATIONS OF LINEAR FUNCTIONALS......Page 154 3.6 WEAK TOPOLOGIES FOR LINEAR SPACES......Page 169 3.7 POLAR SETS AND ANNIHILATORS......Page 173 3.8 EQUICONTINUITY AND C-TOPOLOGIES......Page 178 3.9 THE PRINCIPLE OF UNIFORM BOUNDEDNESS......Page 182 3.10 WEAK TOPOLOGIES FOR NORMED LINEAR SPACES......Page 185 3.11 THE KREIN-MILMAN THEOREM......Page 194 IV GENERAL THEOREMS ON LINEAR OPERATORS......Page 201 4.1 SPACES OF LINEAR OPERATORS......Page 202 4.2 INTEGRAL EQUATIONS OF THE SECOND KIND......Page 209 4.3 L^2 KERNELS......Page 214 4.4 DIFFERENTIAL EQUATIONS AND INTEGRAL EQUATIONS......Page 218 4.5 CLOSED LINEAR OPERATORS......Page 221 4.6 SOME REPRESENTATIONS OF BOUNDED LINEAR OPERATORS......Page 232 4.7 THE M. RIESZ CONVEXITY THEOREM......Page 237 4.8 CONJUGATES OF LINEAR OPERATORS......Page 239 4.9 THEOREMS ABOUT CONTINUOUS INVERSES......Page 247 4.10 THE STATES OF AN OPERATOR AND ITS CONJUGATE......Page 250 4.11 ADJOINT OPERATORS......Page 255 4.12 PROJECTIONS......Page 259 4.13 FREDHOLM OPERATORS......Page 266 V SPECTRAL ANALYSIS OF LINEAR OPERATORS......Page 277 5.1 ANALYTIC VECTOR-VALUED FUNCTIONS......Page 278 5.2 THE RESOLVENT OPERATOR......Page 285 5.3 THE SPECTRUM OF A BOUNDED LINEAR OPERATOR......Page 290 5.4 SUBDIVISIONS OF THE SPECTRUM......Page 295 5.5 REDUCIBILITY......Page 300 5.6 THE ASCENT AND DESCENT OF AN OPERATOR......Page 302 5.7 COMPACT OPERATORS......Page 306 5.8 AN OPERATIONAL CALCULUS......Page 322 5.9 SPECTRAL SETS. THE SPECTRAL MAPPING THEOREM......Page 333 5.10 ISOLATED POINTS OF THE SPECTRUM......Page 341 5.11 OPERATORS WITH A RATIONAL RESOLVENT......Page 349 VI SPECTRAL ANALYSIS IN HILBERT SPACE......Page 354 6.1 BILINEAR AND QUADRATIC FORMS......Page 355 6.2 SYMMETRIC OPERATORS......Page 358 6.3 NORMAL AND SELF-ADJOINT OPERATORS......Page 362 6.4 COMPACT SYMMETRIC OPERATORS......Page 366 6.5 SYMMETRIC OPERATORS WITH COMPACT RESOLVENT......Page 374 6.6 THE SPECTRAL THEOREM FOR BOUNDED SELF-ADJOINT OPERATORS......Page 376 6.7 UNITARY OPERATORS......Page 387 6.8 UNBOUNDED SELF-ADJOINT OPERATORS......Page 393 VII BANACH ALGEBRAS......Page 399 7.1 EXAMPLES OF BANACH ALGEBRAS......Page 400 7.2 SPECTRAL THEORY IN A BANACH ALGEBRA......Page 406 7.3 IDEALS AND HOMOMORPHISMS......Page 413 7.4 COMMUTATIVE BANACH ALGEBRAS......Page 417 7.5 APPLICATIONS AND EXTENSIONS OF THE GELFAND THEORY......Page 428 7.6 B*-ALGEBRAS......Page 439 7.7 THE SPECTRAL THEOREM FOR A NORMAL OPERATOR......Page 443 BIBLIOGRAPHY......Page 458 LIST OF SPECIAL SYMBOLS......Page 468 INDEX......Page 472