BASIC REAL ANALYSIS
CONTENTS I THEORY OF CALCULUS IN ONE REAL VARIABLE 1 REVIEW OF REAL NUMBERS, SEQUENCES, CONTINUITY 2 INTERCHANGE OF LIMITS 3 UNIFORM CONVERGENCE 4 RIEMANN INTEGRAL 5 COMPLEX-VALUED FUNCTIONS 6 TAYLOR’S THEOREM WITH INTEGRAL REMAINDER 7 POWER SERIES AND SPECIAL FUNCTIONS 8 SUMMABILITY 9 WEIERSTRASS APPROXIMATION THEOREM 10 FOURIER SERIES 11 PROBLEMS II METRIC SPACES 1 DE?NITION AND EXAMPLES 2 OPEN SETS AND CLOSED SETS 3 CONTINUOUS FUNCTIONS 4 SEQUENCES AND CONVERGENCE 5 SUBSPACES AND PRODUCTS 6 PROPERTIES OF METRIC SPACES 7 COMPACTNESS AND COMPLETENESS 8 CONNECTEDNESS 9 BAIRE CATEGORY THEOREM 10 PROPERTIES OF C(S) FOR COMPACT METRIC S 11 COMPLETION 12 PROBLEMS III THEORY OF CALCULUS IN SEVERAL REAL VARIABLES 1 OPERATOR NORM 2 NONLINEAR FUNCTIONS AND DIFFERENTIATION 3 VECTOR-VALUED PARTIAL DERIVATIVES AND RIEMANN INTEGRALS 4 EXPONENTIAL OF A MATRIX 5 PARTITIONS OF UNITY 6 INVERSE AND IMPLICIT FUNCTION THEOREMS 7 DE?NITION AND PROPERTIES OF RIEMANN INTEGRAL 8 RIEMANN INTEGRABLE FUNCTIONS 9 FUBINI’S THEOREM FOR THE RIEMANN INTEGRAL 10 CHANGE OF VARIABLES FOR THE RIEMANN INTEGRAL IV THEORY OF ORDINARY DIFFERENTIAL EQUATIONS AND SYSTEMS 1 QUALITATIVE FEATURES AND EXAMPLES 2 EXISTENCE AND UNIQUENESS 3 DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS 4 INTEGRAL CURVES 5 LINEAR EQUATIONS AND SYSTEMS, WRONSKIAN 6 HOMOGENEOUS EQUATIONS WITH CONSTANT COEF?CIENTS 7 HOMOGENEOUS SYSTEMS WITH CONSTANT COEF?CIENTS 8 SERIES SOLUTIONS IN THE SECOND-ORDER LINEAR CASE 9 PROBLEMS V LEBESGUE MEASURE AND ABSTRACT MEASURE THEORY 1 MEASURES AND EXAMPLES 2 MEASURABLE FUNCTIONS 3 LEBESGUE INTEGRAL 4 PROPERTIES OF THE INTEGRAL 5 PROOF OF THE EXTENSION THEOREM 6 COMPLETION OF A MEASURE SPACE 7 FUBINI’S THEOREM FOR THE LEBESGUE INTEGRAL 8 INTEGRATION OF COMPLEX-VALUED AND VECTOR-VALUED FUNCTIONS 9 L1, L2, L?, AND NORMED LINEAR SPACES 10 PROBLEMS VI MEASURE THEORY FOR EUCLIDEAN SPACE 1 LEBESGUE MEASURE AND OTHER BOREL MEASURES 2 CONVOLUTION 3 BOREL MEASURES ON OPEN SETS 4 COMPARISON OF RIEMANN AND LEBESGUE INTEGRALS 5 CHANGE OF VARIABLES FOR THE LEBESGUE INTEGRAL 6 HARDY–LITTLEWOOD MAXIMAL THEOREM 7 FOURIER SERIES AND THE RIESZ–FISCHER THEOREM 8 STIELTJES MEASURES ON THE LINE 9 FOURIER SERIES AND THE DIRICHLET–JORDAN THEOREM 10 DISTRIBUTION FUNCTIONS VII DIFFERENTIATION OF LEBESGUE INTEGRALS ON THE LINE 1 DIFFERENTIATION OF MONOTONE FUNCTIONS 2 ABSOLUTE CONTINUITY, SINGULAR MEASURES, AND LEBESGUE DECOMPOSITION 3 PROBLEMS VIII FOURIER TRANSFORM IN EUCLIDEAN SPACE 1 ELEMENTARY PROPERTIES 2 FOURIER TRANSFORM ON L1, INVERSION FORMULA 3 FOURIER TRANSFORM ON L2, PLANCHEREL FORMULA 4 SCHWARTZ SPACE 5 POISSON SUMMATION FORMULA 6 POISSON INTEGRAL FORMULA 7 HILBERT TRANSFORM 8 PROBLEMS IX L P SPACES 1 INEQUALITIES AND COMPLETENESS 2 CONVOLUTION INVOLVING L P 3 JORDAN AND HAHN DECOMPOSITIONS 4 RADON–NIKODYM THEOREM 5 CONTINUOUS LINEAR FUNCTIONALS ON L P 6 MARCINKIEWICZ INTERPOLATION THEOREM 7 PROBLEMS X TOPOLOGICAL SPACES 1 OPEN SETS AND CONSTRUCTIONS OF TOPOLOGIES 2 PROPERTIES OF TOPOLOGICAL SPACES 3 COMPACTNESS AND LOCAL COMPACTNESS 4 PRODUCT SPACES AND THE TYCHONOFF PRODUCT THEOREM 5 SEQUENCES AND NETS 6 QUOTIENT SPACES 7 URYSOHN’S LEMMA 8 METRIZATION IN THE SEPARABLE CASE 9 ASCOLI–ARZEL`A AND STONE–WEIERSTRASS THEOREMS XI INTEGRATION ON LOCALLY COMPACT SPACES 1 SETTING 2 RIESZ REPRESENTATION THEOREM 3 REGULAR BOREL MEASURES 4 DUAL TO SPACE OF FINITE SIGNED MEASURES 5 PROBLEMS XII HILBERT AND BANACH SPACES 1 DE?NITIONS AND EXAMPLES 2 GEOMETRY OF HILBERT SPACE 3 BOUNDED LINEAR OPERATORS ON HILBERT SPACES 4 HAHN–BANACH THEOREM 5 UNIFORM BOUNDEDNESS THEOREM 6 INTERIOR MAPPING PRINCIPLE