INTEGRATION AND FUNCTIONAL ANALYSIS
CONTENTS PRELIMINARIES SETS SET OPERATIONS NUMBER SYSTEMS RELATIONS FUNCTIONS CARDINALITY ALGEBRAIC STRUCTURES SEMIGROUPS AND GROUPS LINEAR SPACES LINEAR TRANSFORMATIONS QUOTIENT LINEAR SPACES ALGEBRAS METRIC SPACES OPEN AND CLOSED SETS INTERIOR, CLOSURE, AND BOUNDARY SEQUENTIAL CONVERGENCE COMPLETENESS CONTINUITY CATEGORY NORMED LINEAR SPACES NORMS AND SEMINORMS BANACH SPACES COMPLETION OF A NORMED SPACE IN?NITE SERIES IN NORMED SPACES UNORDERED SUMS IN NORMED SPACES BOUNDED LINEAR TRANSFORMATIONS BANACH ALGEBRAS TOPOLOGICAL SPACES NEIGHBORHOOD SYSTEMS NEIGHBORHOOD BASES RELATIVE TOPOLOGY NETS CONTINUITY IN TOPOLOGICAL SPACES DE?NITION AND GENERAL PROPERTIES INITIAL TOPOLOGIES PRODUCT TOPOLOGY FINAL TOPOLOGIES QUOTIENT TOPOLOGY THE SPACE OF CONTINUOUS FUNCTIONS F-SIGMA AND G-DELTA SETS NORMAL TOPOLOGICAL SPACES URYSOHN’S LEMMA TIETZE EXTENSION THEOREM COMPACT TOPOLOGICAL SPACES CONVERGENCE IN COMPACT SPACES COMPACTNESS OF CARTESIAN PRODUCTS CONTINUITY AND COMPACTNESS TOTALLY BOUNDED METRIC SPACES 0 10 EQUICONTINUITY 0 11 THE STONE-WEIERSTRASS THEOREM 0 12 LOCALLY COMPACT TOPOLOGICAL SPACES GENERAL PROPERTIES FUNCTIONS WITH COMPACT SUPPORT FUNCTIONS THAT VANISH AT IN?NITY THE ONE-POINT COMPACTI?CATION 0 13 SPACES OF DI?ERENTIABLE FUNCTIONS 0 14 PARTITIONS OF UNITY 0 15 CONNECTEDNESS I MEASURE AND INTEGRATION MEASURABLE SETS INTRODUCTION MEASURABLE SPACES FIELDS AND SIGMA FIELDS GENERATED SIGMA FIELDS BOREL SETS EXTENDED BOREL SETS PRODUCT SIGMA FIELDS PI-SYSTEMS AND LAMBDA-SYSTEMS EXERCISES MEASURES SET FUNCTIONS PROPERTIES AND EXAMPLES OF MEASURES COMPLETE MEASURE SPACES COMPLETION THEOREM NULL SETS OUTER MEASURE AND MEASURABILITY CONSTRUCTION OF AN OUTER MEASURE CARATH´EODORY’S THEOREM EXTENSION OF A MEASURE THE MEASURE EXTENSION THEOREM APPROXIMATION PROPERTY OF THE EXTENSION COMPLETENESS OF THE EXTENSION UNIQUENESS OF THE EXTENSION LEBESGUE MEASURE THE VOLUME SET FUNCTION CONSTRUCTION OF THE MEASURE LEBESGUE-STIELTJES MEASURES REGULARITY ONE-DIMENSIONAL DISTRIBUTION FUNCTIONS ?HIGHER DIMENSIONAL DISTRIBUTION FUNCTIONS SOME SPECIAL SETS AN UNCOUNTABLE SET WITH LEBESGUE MEASURE ZERO NON-LEBESGUE-MEASURABLE SETS A LEBESGUE MEASURABLE, NON-BOREL SET MEASURABLE FUNCTIONS MEASURABLE TRANSFORMATIONS MEASURABLE NUMERICAL FUNCTIONS CRITERIA FOR MEASURABILITY ALMOST EVERYWHERE PROPERTIES COMBINATORIAL AND LIMIT PROPERTIES OF MEASURABLE FUNCTIONS SIMPLE FUNCTIONS A FUNDAMENTAL CONVERGENCE THEOREM APPLICATIONS CONVERGENCE OF MEASURABLE FUNCTIONS MODES OF CONVERGENCE RELATIONSHIPS AMONG THE MODES OF CONVERGENCE INTEGRATION CONSTRUCTION OF THE INTEGRAL INTEGRAL OF A NONNEGATIVE SIMPLE FUNCTION INTEGRAL OF A REAL-VALUED FUNCTION INTEGRAL OF A COMPLEX-VALUED FUNCTION INTEGRAL OVER A MEASURABLE SET BASIC PROPERTIES OF THE INTEGRAL MONOTONE CONVERGENCE THEOREM LINEARITY OF THE INTEGRAL INTEGRATION AGAINST AN IMAGE MEASURE INTEGRATION AGAINST A MEASURE WITH DENSITY CHANGE OF VARIABLES THEOREM CONNECTIONS WITH THE RIEMANN INTEGRAL ON RD THE DARBOUX INTEGRAL THE RIEMANN INTEGRAL MEASURE ZERO CRITERION FOR RIEMANN INTEGRABILITY IMPROPER RIEMANN INTEGRALS CONVERGENCE THEOREMS THE GENERAL MONOTONE CONVERGENCE THEOREM FATOU’S LEMMA THE DOMINATED CONVERGENCE THEOREM INTEGRATION AGAINST A PRODUCT MEASURE CONSTRUCTION OF THE PRODUCT OF TWO MEASURES FUBINI’S THEOREM THE D-DIMENSIONAL CASE APPLICATIONS OF FUBINI’S THEOREM GAUSSIAN DENSITY INTEGRATION BY PARTS SPHERICAL COORDINATES VOLUME OF A D-DIMENSIONAL BALL INTEGRATION OF RADIAL FUNCTIONS SURFACE AREA OF A D-DIMENSIONAL BALL LP SPACES THE CASE 1 ?P < ? THE CASE P = ? THE CASE 0 < P < ?P-SPACES LP APPROXIMATION APPROXIMATION BY SIMPLE FUNCTIONS APPROXIMATION BY CONTINUOUS FUNCTIONS APPROXIMATION BY STEP FUNCTIONS LP CONVERGENCE UNIFORM INTEGRABILITY CONVEX FUNCTIONS AND JENSEN’S INEQUALITY DI?ERENTIATION SIGNED MEASURES DE?NITION AND A FUNDAMENTAL EXAMPLE THE HAHN-JORDAN DECOMPOSITION COMPLEX MEASURES THE TOTAL VARIATION MEASURE THE VITALI-HAHN-SAKS THEOREM THE BANACH SPACE OF COMPLEX MEASURES INTEGRATION AGAINST A SIGNED OR COMPLEX MEASURE ABSOLUTE CONTINUITY OF MEASURES GENERAL PROPERTIES OF ABSOLUTE CONTINUITY THE RADON-NIKODYM THEOREM LEBESGUE-DECOMPOSITION OF A MEASURE DI?ERENTIATION OF MEASURES DE?NITION AND PROPERTIES OF THE DERIVATIVE CONNECTIONS WITH THE CLASSICAL DERIVATIVE EXISTENCE OF THE MEASURE DERIVATIVE FUNCTIONS OF BOUNDED VARIATION DE?NITION AND BASIC PROPERTIES THE TOTAL VARIATION FUNCTION DI?ERENTIATION OF FUNCTIONS OF BOUNDED VARIATION ABSOLUTELY CONTINUOUS FUNCTIONS FUNDAMENTAL THEOREMS OF CALCULUS FOURIER ANALYSIS ON RD CONVOLUTION OF FUNCTIONS APPROXIMATE IDENTITIES THE FOURIER TRANSFORM THE FOURIER INVERSION THEOREM RAPIDLY DECREASING FUNCTIONS THE PLANCHEREL THEOREM FOURIER ANALYSIS OF MEASURES ON RD CONVOLUTION OF MEASURES THE FOURIER-STIELTJES TRANSFORM MEASURES ON LOCALLY COMPACT SPACES RADON MEASURES CONSEQUENCES OF REGULARITY THE SPACE OF COMPLEX RADON MEASURES THE SUPPORT OF A RADON MEASURE THE RIESZ REPRESENTATION THEOREM PRODUCTS OF RADON MEASURES FINITELY MANY MEASURES IN?NITELY MANY MEASURES VAGUE CONVERGENCE THE DANIELL-STONE REPRESENTATION THEOREM II FUNCTIONAL ANALYSIS GENERAL PROPERTIES OF NORMED SPACES TOPOLOGY AND GEOMETRY SEPARABLE SPACES EQUIVALENT NORMS FINITE DIMENSIONAL SPACES ?STRICTLY CONVEX SPACES THE OPERATOR NORM THE BANACH ALGEBRA B(X) THE DUAL SPACE X? BILINEAR TRANSFORMATIONS CONCRETE REPRESENTATIONS OF DUAL SPACES THE DUAL OF C0 THE DUAL OF C THE DUAL OF LP THE DUAL OF C0(X) SOME CONSTRUCTIONS PRODUCT SPACES DIRECT SUMS QUOTIENT SPACES HAHN-BANACH EXTENSION THEOREMS REAL VERSION COMPLEX VERSION NORMED SPACE VERSION THE BIDUAL OF A NORMED SPACE ?INVARIANT VERSIONS APPLICATIONS OF THE HAHN-BANACH THEOREM THE MOMENT PROBLEM INVARIANT MEANS BANACH LIMITS INVARIANT SET FUNCTIONS BAIRE CATEGORY IN BANACH SPACES THE UNIFORM BOUNDEDNESS PRINCIPLE THE OPEN MAPPING THEOREM THE CLOSED GRAPH THEOREM DIVERGENT FOURIER SERIES VECTOR-VALUED ANALYTIC FUNCTIONS SUMMABILITY SCHAUDER BASES THE DUAL OPERATOR DE?NITION AND PROPERTIES ANNIHILATORS DUALS OF QUOTIENT SPACES AND SUBSPACES 8 10 COMPACT OPERATORS ?FREDHOLM ALTERNATIVE FOR COMPACT OPERATORS LOCALLY CONVEX SPACES GEOMETRY AND TOPOLOGY SEMINORMED SPACES FR´ECHET SPACES CONTINUOUS LINEAR FUNCTIONALS CONTINUITY ON TOPOLOGICAL VECTOR SPACES CONTINUITY ON LOCALLY CONVEX SPACES CONTINUITY ON FINITE DIMENSIONAL SPACES HAHN-BANACH SEPARATION THEOREMS WEAK SEPARATION IN A TVS STRICT SEPARATION IN A LCS SOME CONSEQUENCES OF THE SEPARATION THEOREMS THE BIPOLAR THEOREM STRICT INDUCTIVE LIMITS 10 WEAK TOPOLOGIES ON NORMED SPACES 10 1 THE WEAK TOPOLOGY WEAK SEQUENTIAL CONVERGENCE CONVEXITY AND CLOSURE ?APPLICATION: WEAK BASES 10 2 THE WEAK?TOPOLOGY THE DUAL OF X? W? THE BANACH-ALAOGLU THEOREM ?APPLICATION: MEANS ON FUNCTION SPACES WEAK?CONTINUITY ?THE CLOSED RANGE THEOREM 10 3 RE?EXIVE SPACES EXAMPLES AND BASIC PROPERTIES WEAK COMPACTNESS AND RE?EXIVITY *10 4 UNIFORMLY CONVEX SPACES CONNECTIONS WITH STRICT CONVEXITY WEAK AND STRONG CONVERGENCE CONNECTION WITH RE?EXIVITY 11 HILBERT SPACES 11 1 GENERAL PRINCIPLES SESQUILINEAR FORMS SEMI-INNER-PRODUCT SPACES INNER PRODUCT SPACES HILBERT SPACES ISOMORPHISMS OF HILBERT SPACES 11 2 ORTHOGONALITY ORTHOGONAL COMPLEMENTS 11 3 ORTHONORMAL BASES THE DIMENSION OF A HILBERT SPACE THE GRAM-SCHMIDT PROCESS FOURIER SERIES 11 4 THE HILBERT SPACE ADJOINT BOUNDED SESQUILINEAR FUNCTIONALS THE LAX-MILGRAM THEOREM DE?NITION AND PROPERTIES OF THE ADJOINT B(H) AS A C?-ALGEBRA 12 OPERATOR THEORY 12 1 CLASSES OF OPERATORS NORMAL OPERATORS SELF-ADJOINT OPERATORS POSITIVE OPERATORS ORTHOGONAL PROJECTIONS AND IDEMPOTENTS UNITARY OPERATORS ?PARTIAL ISOMETRIES 12 2 COMPACT OPERATORS AND OPERATORS OF FINITE RANK RANK ONE OPERATORS AN APPROXIMATION THEOREM 12 3 THE SPECTRAL THEOREM FOR COMPACT NORMAL OPERATORS EIGENVALUES AND EIGENVECTORS DIAGONALIZABLE OPERATORS THE SPECTRAL THEOREM *12 4 HILBERT-SCHMIDT OPERATORS THE HILBERT-SCHMIDT NORM THE HILBERT-SCHMIDT INNER PRODUCT THE HILBERT-SCHMIDT OPERATOR A ?B HILBERT-SCHMIDT INTEGRAL OPERATORS *12 5 TRACE CLASS OPERATORS THE TRACE NORM THE TRACE THE DUAL SPACES B0(H)? AND B1(H)? 13 BANACH ALGEBRAS 13 1 INTRODUCTION DE?NITIONS AND EXAMPLES THE GROUP OF INVERTIBLE ELEMENTS THE CAUCHY PRODUCT OF SERIES 13 2 SPECTRAL THEORY THE SPECTRUM OF AN ELEMENT THE SPECTRAL RADIUS FORMULA NORMAL ELEMENTS IN A C?-ALGEBRA 13 3 THE SPECTRUM OF AN ALGEBRA CHARACTERS MAXIMAL IDEALS 13 4 GELFAND THEORY THE REPRESENTATION THEOREM APPLICATION: THE STONE- ?CECH COMPACTI?CATION APPLICATION: WIENER’S THEOREM *13 5 THE NON-UNITAL CASE THE UNITIZATION OF A BANACH ALGEBRA THE NON-UNITAL REPRESENTATION THEOREM THE SPECTRUM OF C0(X) THE SPECTRUM OF L1(RD) 13 6 OPERATOR CALCULUS THE CONTINUOUS FUNCTIONAL CALCULUS APPLICATIONS TO OPERATORS ON HILBERT SPACE THE BOREL FUNCTIONAL CALCULUS THE SPECTRAL THEOREM FOR NORMAL OPERATORS 14 MISCELLANEOUS TOPICS 14 1 WEAK SEQUENTIAL COMPACTNESS THE EBERLEIN-?SMULIAN THEOREM WEAK CONVERGENCE AND UNIFORM INTEGRABILITY THE DUNFORD-PETTIS THEOREM 14 3 CONVEXITY AND COMPACTNESS THE KREIN-?SMULIAN THEOREM MAZUR’S THEOREM THE FINITE DIMENSIONAL CASE 14 4 EXTREME POINTS THE KREIN-MILMAN THEOREM 14 5 APPLICATIONS OF THE KREIN-MILMAN THEOREM EXISTENCE OF ERGODIC MEASURES THE STONE-WEIERSTRASS THEOREM THE BANACH-STONE THEOREM THE LYAPUNOV CONVEXITY THEOREM THE RYLL-NARDZEWSKI FIXED POINT THEOREM 14 6 VECTOR-VALUED INTEGRALS WEAK INTEGRALS IN BANACH SPACES WEAK INTEGRALS IN LOCALLY CONVEX SPACES THE BOCHNER INTEGRAL 14 7 CHOQUET’S THEOREM III APPLICATIONS 15 DISTRIBUTIONS 15 1 GENERAL THEORY THE FR´ECHET SPACE C? K (U) THE SPACES D(U) AND D?(U) EXAMPLES OF DISTRIBUTIONS 15 2 OPERATIONS ON DISTRIBUTIONS DERIVATIVE OF A LOCALLY INTEGRABLE FUNCTION DERIVATIVE OF A DISTRIBUTION MULTIPLICATION BY A SMOOTH FUNCTION COMPOSITION WITH LINEAR MAPS 15 3 DISTRIBUTIONS WITH COMPACT SUPPORT 15 4 CONVOLUTION OF DISTRIBUTIONS 15 5 TEMPERED DISTRIBUTIONS THE FOURIER TRANSFORM OF A TEMPERED DISTRIBUTION 15 6 SOBOLEV THEORY SOBOLEV SPACES APPLICATION: ELLIPTIC PDES SOBOLEV INEQUALITIES 16 ANALYSIS ON LOCALLY COMPACT GROUPS 16 1 TOPOLOGICAL GROUPS DE?NITIONS AND BASIC PROPERTIES TRANSLATION AND UNIFORM CONTINUITY 16 2 HAAR MEASURE EXISTENCE OF HAAR MEASURE ESSENTIAL UNIQUENESS OF HAAR MEASURE THE MODULAR FUNCTION 16 3 SOME CONSTRUCTIONS HAAR MEASURE ON DIRECT PRODUCTS HAAR MEASURE ON SEMIDIRECT PRODUCTS HAAR MEASURE ON QUOTIENT GROUPS 16 4 THE L1-GROUP ALGEBRA CONVOLUTION AND INVOLUTION THE MEASURE ALGEBRA 16 5 REPRESENTATIONS POSITIVE-DE?NITE FUNCTIONS FUNCTIONS OF POSITIVE TYPE UNITARY REPRESENTATIONS IRREDUCIBLE REPRESENTATIONS UNITARY REPRESENTATIONS OF COMPACT GROUPS 16 6 LOCALLY COMPACT ABELIAN GROUPS THE DUAL GROUP BOCHNER’S THEOREM THE INVERSION THEOREM THE PONTRJAGIN DUALITY THEOREM 17 ANALYSIS ON SEMIGROUPS 17 1 SEMIGROUPS WITH TOPOLOGY 17 2 WEAKLY ALMOST PERIODIC FUNCTIONS THE DUAL OF THE SPACE OF WEAKLY ALMOST PERIODIC FUNCTIONS THE WEAKLY ALMOST PERIODIC COMPACTI?CATION INVARIANT MEANS ON WEAKLY ALMOST PERIODIC FUNCTIONS 17 3 ALMOST PERIODIC FUNCTIONS THE ALMOST PERIODIC COMPACTI?CATION 17 4 THE STRUCTURE OF COMPACT SEMIGROUPS ELLIS’S THEOREM EXISTENCE OF IDEMPOTENTS IDEAL STRUCTURE 17 5 STRONGLY ALMOST PERIODIC FUNCTIONS THE STRONGLY ALMOST PERIODIC COMPACTI?CATION 17 6 SEMIGROUPS OF OPERATORS DYNAMICAL PROPERTIES OF SEMIGROUPS OF OPERATORS ERGODIC PROPERTIES OF SEMIGROUPS OF OPERATORS 18 PROBABILITY THEORY 18 1 RANDOM VARIABLES EXPECTATION AND VARIANCE PROBABILITY DISTRIBUTIONS 18 2 INDEPENDENCE INDEPENDENT EVENTS INDEPENDENT RANDOM VARIABLES 18 3 CONDITIONAL EXPECTATION 18 4 SEQUENCES OF INDEPENDENT RANDOM VARIABLES IN?NITE PRODUCT MEASURES THE DISTRIBUTION OF A SEQUENCE OF RANDOM VARIABLES ZERO-ONE LAWS LAWS OF LARGE NUMBERS THE CENTRAL LIMIT THEOREM THE INDIVIDUAL ERGODIC THEOREM STATIONARY PROCESSES 18 5 DISCRETE-TIME MARTINGALES FILTRATIONS DE?NITION AND GENERAL PROPERTIES OF MARTINGALES STOPPING TIMES OPTIONAL SAMPLING UPCROSSINGS CONVERGENCE OF MARTINGALES REVERSED MARTINGALES 18 6 GENERAL STOCHASTIC PROCESSES THE CONSISTENCY CONDITIONS THE PRODUCT OF MEASURABLE SPACES THE KOLMOGOROV EXTENSION THEOREM 18 7 BROWNIAN MOTION CONSTRUCTION OF BROWNIAN MOTION NON-DI?ERENTIABILITY OF BROWNIAN PATHS VARIATION OF BROWNIAN PATHS BROWNIAN MOTION AS A MARTINGALE 18 8 STOCHASTIC INTEGRATION THE ITO INTEGRAL OF A STEP PROCESS THE GENERAL ITO INTEGRAL THE ITO INTEGRAL AS A MARTINGALE 18 9 AN APPLICATION TO FINANCE THE STOCK PRICE PROCESS SELF-FINANCING PORTFOLIOS CALL OPTIONS THE BLACK-SCHOLES OPTION PRICE CHAPTER AJ = A1?A2?