Closed graph theorem

About: Closed graph theorem is a research topic. Over the lifetime, 1010 publications have been published within this topic receiving 24124 citations.

Real and complex analysis

Abstract: Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Sequences and series in Banach spaces

Abstract: I. Riesz's Lemma and Compactness in Banach Spaces. Isomorphic classification of finite dimensional Banach spaces ... Riesz's lemma ... finite dimensionality and compactness of balls ... exercises ... Kottman's separation theorem ... notes and remarks ... bibliography..- II. The Weak and Weak* Topologies: an Introduction. Definition of weak topology ... non-metrizability of weak topology in infinite dimensional Banach spaces ... Mazur's theorem on closure of convex sets ... weakly continuous functional coincide with norm continuous functionals ... the weak* topology ... Goldstine's theorem ... Alaoglu's theorem ... exercises ... notes and remarks ... bibliography..- III. The Eberlein-Smulian Theorem. Weak compactness of closed unit ball is equivalent to reflexivity ... the Eberlein-Smulian theorem ... exercises ... notes and remarks ... bibliography..- IV. The Orlicz-Pettis Theorem. Pettis's measurability theorem ... the Bochner integral ... the equivalence of weak subseries convergence with norm subseries convergence ... exercises ... notes and remarks ... bibliography..- V. Basic Sequences. Definition of Schauder basis ... basic sequences ... criteria for basic sequences ... Mazur's technique for constructing basic sequences ... Pelczynski's proof of the Eberlein-Smulian theorem ... the Bessaga-Pelczynski selection principle... Banach spaces containing co ... weakly unconditionally Cauchy series ... co in dual spaces ... basic sequences spanning complemented subspaces ... exercises ... notes and remarks ... bibliography..- VI. The Dvoretsky-Rogers Theorem. Absolutely P-summing operators ... the Grothendieck-Pietsch domination theorem ... the Dvoretsky-Rogers theorem ... exercises ... notes and remarks ... bibliography..- VII. The Classical Banach Spaces. Weak and pointwise convergence of se-quences in C(?) ... Grothendieck's characterization of weak convergence ... Baire's characterization of functions of the first Baire class ... special features of co, l1l? ... injectivity of l? ... separable injectivity of co ... projectivity of l1 ... l1 is primary ... Pelczynski's decomposition method ... the dual of l? ... the Nikodym-Grothendieck boundedness theorem ... Rosenthal's lemma ... Phillips's lemma ... Schur's theorem ... the Orlicz-Pettis theorem (again)... weak compactness in ca(?) and L1 (?) ... the Vitali-Hahn-Saks theorem ... the Dunford-Pettis theorem ... weak sequential completeness of ca(?) and L1(?) ... the Kadec-Pelczynski theorem ... the Grothendieck-Dieudonne weak compactness criteria in rca ... weak* convergent sequences in l?* are weakly convergent ... Khintchine's Inequalities ... Orlicz's theorem ... unconditionally convergent series in Lp[0, 1], 1 ? p ? 2 ... the Banach-Saks theorem ... Szlenk's theorem ... weakly null sequences in Lp [0, 1], 1 ? p ? 2, have subsequences with norm convergent arithmetic means ... exercises ... notes and remarks ... bibliography..- VIII. Weak Convergence and Unconditionally Convergent Series in Uniformly Convex Spaces. Modulus of convexity ... monotonicity and convexity properties of modulus ... Kadec's theorem on unconditionally convergent series in uniformly convex spaces ... the Milman-Pettis theorem on reflexivity of uniformly convex spaces ... Kakutani's proof that uniformly convex spaces have the Banach-Saks property ... the Gurarii-Gurarii theorem on lp estimates for basic sequences in uniformly convex spaces ... exercises ... notes and remarks ... bibliography..- IX. Extremal Tests for Weak Convergence of Sequences and Series. The Krein-Milman theorem ... integral representations ... Bauer's characterization of extreme points ... Milman's converse to the Krein-Milman theorem ... the Choquet integral representation theorem ... Rainwater's theorem ... the Super lemma ... Namioka's density theorems ... points of weak*-norm continuity of identity map ... the Bessaga-Pelczynski characterization of separable duals ... Haydon's separable generation theorem ... the remarkable renorming procedure of Fonf ... Elton's extremal characterization of spaces without co-subspaces ... exercises ... notes and remarks ... bibliography..- X. Grothendieck's Inequality and the Grothendieck-Lindenstrauss-Pelczynski Cycle of Ideas. Rietz's proof of Grothendieck's inequality ... definition of ?p spaces ... every operator from a ?1-space to a ?2-space is absolutely 1-summing ... every operator from a L? space to ?1 space is absolutely 2-summing ... c0, l1 and l2 have unique unconditional bases ... exercises ... notes and remarks ... bibliography..- An Intermission: Ramsey's Theorem. Mathematical sociology ... completely Ramsey sets ... Nash-Williams' theorem ... the Galvin-Prikry theorem ... sets with the Baire property ... notes and remarks ... bibliography..- XI. Rosenthal's l1-theorem. Rademacher-like systems ... trees ... Rosenthal's I1-theorem ... exercises ... notes and remarks ... bibliography..- XII. The Josefson-Nissenzweig Theorem. Conditions insuring l1's presence in a space given its presence in the dual ... existence of weak* null sequences of norm-one functionals ... exercises ... notes and remarks ... bibliography..- XIII. Banach Spaces with Weak*-Sequentially Compact Dual Balls. Separable Banach spaces have weak* sequentially compact dual balls ... stability results ... Grothendieck's approximation criteria for relative weak compactness ... the Davis-Figiel-Johnson-Pelczynski scheme ... Amir-Lindenstrauss theorem ... subspaces of weakly compactly generated spaces have weak* sequentially compact dual balls ... so do spaces each of whose separable subspaces have a separable dual, thanks to Hagler and Johnson ... the Odell-Rosenthal characterization of separable spaces with weak* sequentially compact second dual balls ... exercises ... notes and remarks ... bibliography..- XIV. The Elton-Odell (l + ?)Separation Theorem. James's co-distortion theorem ... Johnson's combinatorial designs for detecting co's presence ... the Elton-Odell proof that each infinite dimensional Banach space contains a (l + ?)-separated sequence of norm-one elements ... exercises ... notes and remarks . . bibliography..

Lectures on closed geodesics

Abstract: 1. The Hilbert Manifold of Closed Curves.- 1.1 Hilbert Manifolds.- 1.2 The Manifold of Closed Curves.- 1.3 Riemannian Metric and Energy Integral of the Manifold of Closed Curves.- 1.4 The Condition (C) of Palais and Smale and its Consequences.- 2. The Morse-Lusternik-Schnirelmann Theory on the Manifold of Closed Curves.- 2.1 The Lusternik-Schnirelmann Theory on ?M.- 2.2 The Space of Unparameterized Closed Curves.- 2.3 Closed Geodesics on Spheres.- 2.4 Morse Theory on ?M.- 2.5 The Morse Complex.- 3. The Geodesic Flow.- 3.1 Hamiltonian Systems.- 3.2 The Index Theorem for Closed Geodesics.- 3.3 Properties of the Poincare Map.- 3.3 Appendix. The Birkhoff-Lewis Fixed Point Theorem. By J. Moser.- 4. On the Existence of Many Closed Geodesics.- 4.1 Critical Points in ?M and the Theorem of Fet.- 4.2 The Theorem of Gromoll-Meyer.- 4.3 The Existence of Infinitely Many Closed Geodesics.- 4.3 Appendix. The Minimal Model for the Rational Homotopy Type of ?M. By J. Sacks.- 4.4 Some Generic Existence Theorems.- 5. Miscellaneous Results.- 5.1 The Theorem of the Three Closed Geodesics.- 5.2 Some Special Manifolds of Elliptic Type.- 5.3 Geodesics on Manifolds of Hyperbolic and Parabolic Type.- Appendix. The Theorem of Lusternik and Schnirelmann.- A.2 Closed Curves without Self-intersections on the 2-sphere.- A.3 The Theorem of Lusternik and Schnirelmann.