INFINITE DIMENSIONAL ANALYSIS
CONTENTS ODDS AND ENDS NUMBERS SETS RELATIONS, CORRESPONDENCES, AND FUNCTIONS A BESTIARY OF RELATIONS EQUIVALENCE RELATIONS ORDERS AND SUCH REAL FUNCTIONS DUALITY OF EVALUATION IN?NIT\/YIES THE DIAGONAL THEOREM AND RUSSELL’S PARADOX THE AXIOM OF CHOICE AND AXIOMATIC SET THEORY ZORN’S LEMMA ORDINALS TOPOLOGY TOPOLOGICAL SPACES NEIGHBORHOODS AND CLOSURES DENSE SUBSETS NETS FILTERS NETS AND FILTERS CONTINUOUS FUNCTIONS COMPACTNESS NETS VS SEQUENCES SEMICONTINUOUS FUNCTIONS SEPARATION PROPERTIES COMPARING TOPOLOGIES WEAK TOPOLOGIES THE PRODUCT TOPOLOGY POINTWISE AND UNIFORM CONVERGENCE LOCALLY COMPACT SPACES THE STONE– ?CECH COMPACTI?CATION STONE– ?CECH COMPACTI?CATION OF A DISCRETE SET PARACOMPACT SPACES AND PARTITIONS OF UNITY METRIZABLE SPACES METRIC SPACES COMPLETENESS UNIFORMLY CONTINUOUS FUNCTIONS SEMICONTINUOUS FUNCTIONS ON METRIC SPACES DISTANCE FUNCTIONS EMBEDDINGS AND COMPLETIONS COMPACTNESS AND COMPLETENESS COUNTABLE PRODUCTS OF METRIC SPACES THE HILBERT CUBE AND METRIZATION LOCALLY COMPACT METRIZABLE SPACES THE BAIRE CATEGORY THEOREM CONTRACTION MAPPINGS THE CANTOR SET THE BAIRE SPACE NN UNIFORMITIES THE HAUSDOR?DISTANCE THE HAUSDOR?METRIC TOPOLOGY TOPOLOGIES FOR SPACES OF SUBSETS THE SPACE C(X, Y) MEASURABILITY ALGEBRAS OF SETS RINGS AND SEMIRINGS OF SETS DYNKIN’S LEMMA THE BOREL ?-ALGEBRA MEASURABLE FUNCTIONS THE SPACE OF MEASURABLE FUNCTIONS SIMPLE FUNCTIONS THE ?-ALGEBRA INDUCED BY A FUNCTION PRODUCT STRUCTURES CARATHÉODORY FUNCTIONS BOREL FUNCTIONS AND CONTINUITY THE BAIRE ?-ALGEBRA TOPOLOGICAL VECTOR SPACES LINEAR TOPOLOGIES ABSORBING AND CIRCLED SETS METRIZABLE TOPOLOGICAL VECTOR SPACES THE OPEN MAPPING AND CLOSED GRAPH THEOREMS FINITE DIMENSIONAL TOPOLOGICAL VECTOR SPACES CONVEX SETS CONVEX AND CONCAVE FUNCTIONS SUBLINEAR FUNCTIONS AND GAUGES THE HAHN–BANACH EXTENSION THEOREM SEPARATING HYPERPLANE THEOREMS SEPARATION BY CONTINUOUS FUNCTIONALS LOCALLY CONVEX SPACES AND SEMINORMS SEPARATION IN LOCALLY CONVEX SPACES DUAL PAIRS TOPOLOGIES CONSISTENT WITH A GIVEN DUAL POLARS S-TOPOLOGIES THE MACKEY TOPOLOGY THE STRONG TOPOLOGY NORMED SPACES NORMED AND BANACH SPACES LINEAR OPERATORS ON NORMED SPACES THE NORM DUAL OF A NORMED SPACE THE UNIFORM BOUNDEDNESS PRINCIPLE WEAK TOPOLOGIES ON NORMED SPACES METRIZABILITY OF WEAK TOPOLOGIES CONTINUITY OF THE EVALUATION ADJOINT OPERATORS PROJECTIONS AND THE ?XED SPACE OF AN OPERATOR HILBERT SPACES CONVEXITY EXTENDED-VALUED CONVEX FUNCTIONS LOWER SEMICONTINUOUS CONVEX FUNCTIONS SUPPORT POINTS SUBGRADIENTS SUPPORTING HYPERPLANES AND CONES CONVEX FUNCTIONS ON ?NITE DIMENSIONAL SPACES SEPARATION AND SUPPORT IN ?NITE DIMENSIONAL SPACES SUPPORTING CONVEX SUBSETS OF HILBERT SPACES THE BISHOP–PHELPS THEOREM SUPPORT FUNCTIONALS SUPPORT FUNCTIONALS AND THE HAUSDOR?METRIC EXTREME POINTS OF CONVEX SETS QUASICONVEXITY POLYTOPES AND WEAK NEIGHBORHOODS EXPOSED POINTS OF CONVEX SETS RIESZ SPACES ORDERS, LATTICES, AND CONES ORDER BOUNDED SETS ORDER AND LATTICE PROPERTIES THE RIESZ DECOMPOSITION PROPERTY DISJOINTNESS RIESZ SUBSPACES AND IDEALS ORDER CONVERGENCE AND ORDER CONTINUITY BANDS POSITIVE FUNCTIONALS EXTENDING POSITIVE FUNCTIONALS POSITIVE OPERATORS TOPOLOGICAL RIESZ SPACES THE BAND GENERATED BY E? RIESZ PAIRS SYMMETRIC RIESZ PAIRS BANACH LATTICES FRÉCHET AND BANACH LATTICES THE STONE–WEIERSTRASS THEOREM LATTICE HOMOMORPHISMS AND ISOMETRIES ORDER CONTINUOUS NORMS AM- AND AL-SPACES THE INTERIOR OF THE POSITIVE CONE POSITIVE PROJECTIONS THE CURIOUS AL-SPACE BV0 10 CHARGES AND MEASURES SET FUNCTIONS LIMITS OF SEQUENCES OF MEASURES OUTER MEASURES AND MEASURABLE SETS THE CARATHÉODORY EXTENSION OF A MEASURE MEASURE SPACES LEBESGUE MEASURE PRODUCT MEASURES MEASURES ON RN ATOMS 10 10 THE AL-SPACE OF CHARGES 10 11 THE AL-SPACE OF MEASURES 10 12 ABSOLUTE CONTINUITY 11 INTEGRALS THE INTEGRAL OF A STEP FUNCTION FINITELY ADDITIVE INTEGRATION OF BOUNDED FUNCTIONS THE LEBESGUE INTEGRAL CONTINUITY PROPERTIES OF THE LEBESGUE INTEGRAL THE EXTENDED LEBESGUE INTEGRAL ITERATED INTEGRALS THE RIEMANN INTEGRAL THE BOCHNER INTEGRAL THE GELFAND INTEGRAL 11 10 THE DUNFORD AND PETTIS INTEGRALS 12 MEASURES AND TOPOLOGY BOREL MEASURES AND REGULARITY REGULAR BOREL MEASURES THE SUPPORT OF A MEASURE NONATOMIC BOREL MEASURES ANALYTIC SETS THE CHOQUET CAPACITY THEOREM 13 LP-SPACES LP-NORMS INEQUALITIES OF HÖLDER AND MINKOWSKI DENSE SUBSPACES OF LP-SPACES SUBLATTICES OF LP-SPACES SEPARABLE L1-SPACES AND MEASURES THE RADON–NIKODYM THEOREM EQUIVALENT MEASURES DUALS OF LP-SPACES LYAPUNOV’S CONVEXITY THEOREM 13 10 CONVERGENCE IN MEASURE 13 11 CONVERGENCE IN MEASURE IN LP-SPACES 13 12 CHANGE OF VARIABLES 14 RIESZ REPRESENTATION THEOREMS THE AM-SPACE BB(?) AND ITS DUAL THE DUAL OF CB(X) FOR NORMAL SPACES THE DUAL OF CC(X) FOR LOCALLY COMPACT SPACES BAIRE VS BOREL MEASURES HOMOMORPHISMS BETWEEN C(X)-SPACES 15 PROBABILITY MEASURES THE WEAK* TOPOLOGY ON P (X) EMBEDDING X IN P (X) PROPERTIES OF P (X) THE MANY FACES OF P (X) COMPACTNESS IN P (X) THE KOLMOGOROV EXTENSION THEOREM 16 SPACES OF SEQUENCES THE BASIC SEQUENCE SPACES THE SEQUENCE SPACES RN AND ? THE SEQUENCE SPACE C0 THE SEQUENCE SPACE C THE ?P-SPACES ?1 AND THE SYMMETRIC RIESZ PAIR ???, ?1? THE SEQUENCE SPACE ?? MORE ON ?? ?= BA(N) EMBEDDING SEQUENCE SPACES 16 10 BANACH–MAZUR LIMITS AND INVARIANT MEASURES 16 11 SEQUENCES OF VECTOR SPACES 17 CORRESPONDENCES BASIC DE?NITIONS CONTINUITY OF CORRESPONDENCES HEMICONTINUITY AND NETS OPERATIONS ON CORRESPONDENCES THE MAXIMUM THEOREM VECTOR-VALUED CORRESPONDENCES DEMICONTINUOUS CORRESPONDENCES KNASTER–KURATOWSKI–MAZURKIEWICZ MAPPINGS FIXED POINT THEOREMS 17 10 CONTRACTION CORRESPONDENCES 17 11 CONTINUOUS SELECTORS 18 MEASURABLE CORRESPONDENCES MEASURABILITY NOTIONS COMPACT-VALUED CORRESPONDENCES AS FUNCTIONS MEASURABLE SELECTORS CORRESPONDENCES WITH MEASURABLE GRAPH CORRESPONDENCES WITH COMPACT CONVEX VALUES INTEGRATION OF CORRESPONDENCES 19 MARKOV TRANSITIONS MARKOV AND STOCHASTIC OPERATORS MARKOV TRANSITIONS AND KERNELS CONTINUOUS MARKOV TRANSITIONS INVARIANT MEASURES ERGODIC MEASURES MARKOV TRANSITION CORRESPONDENCES RANDOM FUNCTIONS DILATIONS MORE ON MARKOV OPERATORS 19 10 A NOTE ON DYNAMICAL SYSTEMS 20 ERGODICITY MEASURE-PRESERVING TRANSFORMATIONS AND ERGODICITY BIRKHO?’S ERGODIC THEOREM ERGODIC OPERATORS REFERENCES