HISTORY OF BANACH SPACES AND LINEAR OPERATORS

By U. KUARN

CONTENTS 1 THE BIRTH OF BANACH SPACES 11 COMPLETE NORMED LINEAR SPACES 12 LINEAR SPACES 13 METRIC SPACES 14 MINKOWSKI SPACES 15 HILBERT SPACES 16 ALBERT A BENNETT AND KENNETH W LAMSON 17 NORBERT WIENER 18 EDUARD HELLY AND HANS HAHN 19 SUMMARY 2 HISTORICAL ROOTS AND BASIC RESULTS 21 OPERATORS 22 FUNCTIONALS AND DUAL OPERATORS 23 THE MOMENT PROBLEM AND THE HAHN–BANACH THEOREM 24 THE UNIFORM BOUNDEDNESS PRINCIPLE 25 THE CLOSED GRAPH THEOREM AND THE OPEN MAPPING THEOREM 26 RIESZ–SCHAUDER THEORY COMPLETELY CONTINUOUS OPERATORS FINITE RANK OPERATORS APPROXIMABLE OPERATORS COMPACT OPERATORS RESOLVENTS, SPECTRA, AND EIGENVALUES CLASSICAL OPERATOR IDEALS 27 BANACH’S MONOGRAPH 3 TOPOLOGICAL CONCEPTS – WEAK TOPOLOGIES 31 WEAKLY CONVERGENT SEQUENCES 32 TOPOLOGICAL SPACES AND TOPOLOGICAL LINEAR SPACES TOPOLOGICAL SPACES NETS AND ?LTERS COMPACTNESS TOPOLOGICAL LINEAR SPACES LOCALLY BOUNDED LINEAR SPACES 33 LOCALLY CONVEX LINEAR SPACES AND DUALITY LOCALLY CONVEX LINEAR SPACES WEAK TOPOLOGIES AND DUAL SYSTEMS SEPARATION OF CONVEX SETS TOPOLOGIES ON L(X,Y) 34 WEAK?AND WEAK COMPACTNESS TYCHONOFF’S THEOREM WEAK?COMPACTNESS THEOREM WEAK COMPACTNESS AND RE?EXIVITY 35 WEAK SEQUENTIAL COMPLETENESS AND THE SCHUR PROPERTY 36 TRANS?NITELY CLOSED SETS 4 CLASSICAL BANACH SPACES 41 BANACH LATTICES 42 MEASURES AND INTEGRALS ON ABSTRACT SETS SET-THEORETIC OPERATIONS MEASURES FROM MEASURES TO INTEGRALS INTEGRALS AND THE BANACH SPACES L1 BANACH SPACES OF ADDITIVE SET FUNCTIONS 43 THE DUALITY BETWEEN L1 AND L? 44 THE BANACH SPACES LP 45 BANACH SPACES OF CONTINUOUS FUNCTIONS 46 MEASURES AND INTEGRALS ON TOPOLOGICAL SPACES 47 MEASURES VERSUS INTEGRALS 48 ABSTRACT LP- AND M-SPACES BOOLEAN ALGEBRAS MEASURE ALGEBRAS ABSTRACT LP-SPACES ABSTRACT M-SPACES THE DUNFORD–PETTIS PROPERTY 49 STRUCTURE THEORY ISOMORPHISMS, INJECTIONS, SURJECTIONS, AND PROJECTIONS EXTENSIONS AND LIFTINGS ISOMETRIC AND ISOMORPHIC CLASSI?CATION 410 OPERATOR IDEALS AND OPERATOR ALGEBRAS SCHATTEN–VON NEUMANN IDEALS BANACH ALGEBRAS B?-ALGEBRAS = C?-ALGEBRAS W ?-ALGEBRAS 411 COMPLEXI?CATION 5 BASIC RESULTS FROM THE POST-BANACH PERIOD 51 ANALYSIS IN BANACH SPACES CONVERGENCE OF SERIES INTEGRATION OF VECTOR-VALUED FUNCTIONS REPRESENTATION OF OPERATORS FROM L1 INTO X THE RADON–NIKODYM PROPERTY: ANALYTIC ASPECTS REPRESENTATION OF OPERATORS FROM LP INTO X REPRESENTATION OF OPERATORS FROM C(K) INTO X VECTOR-VALUED ANALYTIC FUNCTIONS ON THE COMPLEX PLANE GˆATEAUX AND FR´ECHET DERIVATIVES POLYNOMIALS AND DERIVATIVES OF HIGHER ORDER ANALYTIC FUNCTIONS ON BANACH SPACES 52 SPECTRAL THEORY OPERATIONAL CALCULUS FREDHOLM OPERATORS RIESZ OPERATORS INVARIANT SUBSPACES SPECTRAL OPERATORS 53 SEMI-GROUPS OF OPERATORS DETERMINISTIC AND STOCHASTIC PROCESSES THE HILLE–YOSIDA THEOREM ANALYTIC SEMI-GROUPS THE ABSTRACT CAUCHY PROBLEM ERGODIC THEORY 54 CONVEXITY, EXTREME POINTS, AND RELATED TOPICS THE KRE?IN–MILMAN THEOREM INTEGRAL REPRESENTATIONS GELFAND–NA?IMARK–SEGAL REPRESENTATIONS THE RADON–NIKODYM PROPERTY: GEOMETRIC ASPECTS CONVEX AND CONCAVE FUNCTIONS LYAPUNOV’S THEOREM AND THE BANG-BANG PRINCIPLE 55 GEOMETRY OF THE UNIT BALL STRICT CONVEXITY AND SMOOTHNESS UNIFORM CONVEXITY AND UNIFORM SMOOTHNESS FURTHER CONCEPTS RELATED TO CONVEXITY AND SMOOTHNESS APPLICATIONS OF CONVEXITY AND SMOOTHNESS COMPLEX CONVEXITY 56 BASES SCHAUDER BASES AND BASIC SEQUENCES BIORTHOGONAL SYSTEMS BASES AND STRUCTURE THEORY BASES IN CONCRETE BANACH SPACES UNCONDITIONAL BASIC SEQUENCES IN HILBERT SPACES WAVELETS SCHAUDER DECOMPOSITIONS 57 TENSOR PRODUCTS AND APPROXIMATION PROPERTIES BILINEAR MAPPINGS TENSOR PRODUCTS NUCLEAR AND INTEGRAL OPERATORS APPROXIMATION PROPERTIES 6 MODERN BANACH SPACE THEORY – SELECTED TOPICS 61 GEOMETRY OF BANACH SPACES BANACH–MAZUR DISTANCE, PROJECTION, AND BASIS CONSTANTS DVORETZKY’S THEOREM FINITE REPRESENTABILITY, ULTRAPRODUCTS, AND SPREADING MODELS LP-SPACES LOCAL UNCONDITIONAL STRUCTURE BANACH SPACES CONTAINING LNP’S UNIFORMLY RADEMACHER TYPE AND COTYPE, GAUSS TYPE AND COTYPE FOURIER TYPE AND COTYPE, WALSH TYPE AND COTYPE SUPERRE?EXIVITY, HAAR TYPE AND COTYPE UMD SPACES = HT SPACES VOLUME RATIOS AND GROTHENDIECK NUMBERS 62 S-NUMBERS S-NUMBERS OF OPERATORS ON HILBERT SPACE AXIOMATICS OF S-NUMBERS EXAMPLES OF S-NUMBERS ENTROPY NUMBERS S-NUMBERS OF DIAGONAL OPERATORS S-NUMBERS VERSUS WIDTHS 63 OPERATOR IDEALS IDEALS ON HILBERT SPACE BASIC CONCEPTS OF IDEAL THEORY ON BANACH SPACES IDEALS ASSOCIATED WITH S-NUMBERS LOCAL THEORY OF QUASI-BANACH IDEALS AND TRACE DUALITY P-FACTORABLE OPERATORS P-SUMMING OPERATORS P-NUCLEAR AND P-INTEGRAL OPERATORS SPECI?C COMPONENTS OF OPERATOR IDEALS GROTHENDIECK’S THEOREM LIMIT ORDER OF OPERATOR IDEALS BANACH IDEALS AND TENSOR PRODUCTS IDEAL NORMS COMPUTED WITH N VECTORS OPERATOR IDEALS AND CLASSES OF BANACH SPACES SUPER WEAKLY COMPACT OPERATORS, HAAR TYPE AND COTYPE UMD OPERATORS AND HT OPERATORS RADON–NIKODYM PROPERTY: OPERATOR-THEORETIC ASPECTS IDEAL NORMS AND PARAMETERS OF MINKOWSKI SPACES IDEAL NORMS OF ?NITE RANK OPERATORS OPERATOR IDEALS AND CLASSES OF LOCALLY CONVEX LINEAR SPACES 64 EIGENVALUE DISTRIBUTIONS EIGENVALUE SEQUENCES AND CLASSICAL RESULTS INEQUALITIES BETWEEN S-NUMBERS AND EIGENVALUES EIGENVALUES OF P-SUMMING OPERATORS EIGENVALUES OF NUCLEAR OPERATORS OPERATORS OF EIGENVALUE TYPE LP,Q 65 TRACES AND DETERMINANTS TRACES FREDHOLM DENOMINATORS AND DETERMINANTS REGULARIZED FREDHOLM DENOMINATORS THE GOHBERG–GOLDBERG–KRUPNIK APPROACH EIGENVALUES AND ZEROS OF ENTIRE FUNCTIONS COMPLETENESS OF ROOT VECTORS DETERMINANTS: PROS AND CONS 66 INTERPOLATION THEORY CLASSICAL CONVEXITY THEOREMS INTERPOLATION METHODS COMPLEX AND REAL INTERPOLATION METHODS LORENTZ SPACES APPLICATIONS OF INTERPOLATION THEORY INTERPOLATION OF OPERATOR IDEALS NEW TRENDS IN INTERPOLATION THEORY 67 FUNCTION SPACES H¨OLDER–LIPSCHITZ SPACES SOBOLEV SPACES BESOV SPACES LIZORKIN–TRIEBEL SPACES INTERPOLATION OF FUNCTION SPACES SPACES OF SMOOTH FUNCTIONS: SUPPLEMENTS BASES OF BESOV SPACES EMBEDDING OPERATORS SPACES OF VECTOR-VALUED FUNCTIONS INTEGRAL OPERATORS DIFFERENTIAL OPERATORS HARDY SPACES BERGMAN SPACES ORLICZ SPACES 68 PROBABILITY THEORY ON BANACH SPACES BAIRE, BOREL, AND RADON MEASURES CYLINDRICAL MEASURES CHARACTERISTIC FUNCTIONALS RADONIFYING OPERATORS GAUSSIAN MEASURES WIENER MEASURE VECTOR-VALUED RANDOM VARIABLES RADEMACHER SERIES THE LAW OF LARGE NUMBERS THE CENTRAL LIMIT THEOREM VECTOR-VALUED MARTINGALES 69 FURTHER TOPICS TOPOLOGICAL PROPERTIES OF BANACH SPACES TOPOLOGICAL CLASSI?CATION OF BANACH SPACES THE ANALYTIC RADON–NIKODYM PROPERTY M-IDEALS STABLE BANACH SPACES THREE SPACE PROPERTIES FUNCTORS IN CATEGORIES OF BANACH SPACES LOCAL SPECTRAL THEORY HANKEL AND TOEPLITZ OPERATORS COMPOSITION OPERATORS METHODS OF SUMMABILITY LACUNARITY THE LINEAR GROUP OF A BANACH SPACE MANIFOLDS MODELED ON BANACH SPACES ASYMPTOTIC GEOMETRIC ANALYSIS OPERATOR SPACES OMISSIONS 7 MISCELLANEOUS TOPICS 71 BANACH SPACE THEORY AS A PART OF MATHEMATICS 72 SPACES VERSUS OPERATORS 73 MODERN TECHNIQUES OF BANACH SPACE THEORY PROBABILISTIC METHODS COMBINATORIAL METHODS 74 COUNTEREXAMPLES A SELECTION OF TYPICAL COUNTEREXAMPLES SPACES WITHOUT THE APPROXIMATION PROPERTY TSIRELSON’S SPACE THE DISTORTION PROBLEM THE GOWERS–MAUREY STORY A FEW COUNTEREXAMPLES OF OPERATOR THEORY 75 BANACH SPACES AND AXIOMATIC SET THEORY 8 MATHEMATICS IS MADE BY MATHEMATICIANS 81 VICTIMS OF POLITICS 82 SCIENTI?C SCHOOLS IN BANACH SPACE THEORY POLAND USA THE FORMER SOVIET UNION, MAINLY RUSSIA AND UKRAINE ISRAEL FRANCE UNITED KINGDOM FURTHER COUNTRIES GERMANY 83 SHORT BIOGRAPHIES OF SOME FAMOUS MATHEMATICIANS FRIGYES RIESZ EDUARD HELLY STEFAN BANACH JOHN VON NEUMANN MARK GRIGORIEVICH KRE?IN ALEXANDRE GROTHENDIECK NICOLAS BOURBAKI 84 BANACH SPACE THEORY AT THE ICMS 85 THE BANACH SPACE ARCHIVE