Riassunto
The extraordinarily rapid advances made in mathematics since World War II have resulted in analysis becoming an enormous organism spread ing in all directions. Gone for good surely are the days of the great French "courses of analysis" which embodied the whole of the "ana lytical" knowledge of the times in three volumes-as the classical work of Camille Jordan. Perhaps that is why present-day textbooks of anal ysis are disproportionately modest relative to the present state of the art. More: they have "retreated" to the state before Jordan and Goursat. In recent years the scene has been changing rapidly: Jean Dieudon ne is offering us his monumentel Elements d'Analyse (10 volumes) written in the spirit of the great French Course d'Analyse. To the best of my knowledge, the present book is the only one of its size: starting from scratch-from rational numbers, to be precise-it goes on to the theory of distributions, direct integrals, analysis on com plex manifolds, Kahler manifolds, the theory of sheaves and vector bun dles, etc. My objective has been to show the young reader the beauty and wealth of the unsual world of modern mathematical analysis and to show that it has its roots in the great mathematics of the 19th century and mathematical physics. I do know that the young mind eagerly drinks in beautiful and difficult things, rejoicing in the fact that the world is great and teeming with adventure.
Contenuti
XII. Topology. Uniform Structures. Function Spaces.- 1. Topological Spaces.- 2. A Basis of Neighbourhoods. Axioms of Countability.- 3. Filters.- 4. Compact Spaces.- 5. The Cartesian Product of Topological Spaces.- 6. Metric Spaces. Baire Spaces.- 7. The Topological Product of Metric Spaces.- 8. Semicontinuous Functions.- 9. Regular Spaces.- 10. Uniform Spaces. The Completeness of a Space.- 11. Precompact and Compact Uniform Spaces.- 12. Uniform Structures on Spaces of Mappings.- 13. Families of Equicontinuous Mappings. General Ascoli Theorem.- 14. Complements and Exercises.- XIII. Theory of the Integral.- 1. Compactification of the Real Line.- 2. The Daniel—Stone Integral.- 3. The Functional ?* and Its Properties.- 4. The Outer Measure.- 5. Seminorms Np. The Minkowski and Hölder Inequalities.- 6. The Spaces ?p.- 7. The Spaces ?p.- 8. The Space ?1 of Integrable Functions. The Integral.- 9. The Set ? for the Radon Integral. Semicontinuity.- 10. Application of the Lebesgue Theorem. Integrals with a Parameter. Integration of Series.- 11. Measurable Functions.- 12. Measure. Integrable Sets.- 13. The Stone Axiom and Its Consequences.- 14. The Spaces Lp.- 15. The Hahn-Banach Theorem.- 16. Hilbert Spaces. Theorem on Orthogonal Decomposition. The General Form of a Linear Functional.- 17. The Strong Stone Axiom and Its Consequences.- 18. The Tensor Product of Integrals.- 19. The Radon Integral. Stone’s Second Procedure.- 20. Finite Radon Measures. Tough Measures.- 21. The Tensor Product of Radon Integrals.- 22. The Lebesgue Integral on Rn. Change of Variables.- 23. Mapping of Radon Integrals.- 24. Integrals with Density. The Radon-Nikodym Theorem.- 25. The Wiener Integral.- 26. The Kolmogorov Theorem.- 27. Integration of Vector Fields.- 28. Direct Integrals of Hilbert Spaces.- 29. On the Equivalence of the Stone and Radon Integral Theories.- 30. From Measure to Integral.- XIV. Tensor Analysis. Harmonic Forms. Cohomology. Applications to Electrodynamics.- 1. Alternating Maps. Grassmann Algebra.- 2. Differential Forms.- 3. Cohomology Spaces. Poincaré Lemma.- 4. Integration of Differential Forms.- 5. Elements of Vector Analysis.- 6. The Differentiate Manifold.- 7. Tangent Spaces.- 8. Covariant Tensor Fields. Riemannian Metric and Differential Forms on a Manifold.- 9. Orientation of Manifolds. Examples.- 10. Poincaré—Stokes Theorem for a Manifold with Boundary.- 11. Tensor Densities. Weyl Duality. Homology.- 12. Weyl Duality and Hodge *operator. Generalized Green’s formulas on Riemannian Manifold.- 13. Harmonic Forms. Hodge-Kodaira-de Rham Theory.- 14. Application to Electrodynamics.- 15. Invariant Forms (Hurwitz Integral). Cohomology of Compact Lie Groups.- 16. Complements. Exercises.- XV. Elementary Properties of Holomorphic Functions of Several Variables. Harmonic Functions.- 1. Holomorphic Mappings. Cauchy—Riemann Equations.- 2. Differential Forms on Complex Manifolds. Forms of Type (p, q). Operators d? and d??.- 3. Cauchy’s Formula and its Applications.- 4. The Topology of the Space of Holomorphic Functions A(?).- 5. Elementary Properties of Harmonic Functions.- 6. Green’s Function. Poisson Integral Formula. Harnack Theorems..- 7. Subharmonic Functions. Perron’s Solution of the Dirichlet Proble.- XVI. Complex Analysis in One Dimension (Riemann Surfaces.- 1. Zeros of Holomorphic Functions of One Variable.- 2. Functions Holomorphic in an Annulus. The Laurent Expansion. Singularities.- 3. Meromorphic Functions.- 4. Application of the Calculus of Residues to the Evaluation of Integrals.- 5. Applications of the Argument Principle.- 6. Functions and Differential Forms on Riemann Surfaces.- 7. Analytic Continuation. Coverings. Fundamental Group. The Theory of Poincaré.- 8. The Koebe—Riemann Theorem. Non-Euclidean Geometry. Möbius Transformations.- 9. The Perron Method for Riemann Surfaces. The Radó Theorem.- 10. Resolutive Functions. Harmonic Measures. Brelot’s Theorem.- 11. The Green’s Function of a Riemann Surface.- 12. The Uniformization Theorem.- 13. Runge’s Theorem. Theorem of Behnke and Stein. Theorem of Malgrange.- 14. Cousin Problems for Open Riemann Surfaces. Theorems of Mittag-Leffler and Weierstrass.- 15. Examples of Partial Fractions and Factorizations. Functions cos ?z, ?2/sin2 ?z, ?(z). Mellin and Hankel Formulae. Canonical Products.- 16. Elliptic Functions. Eisenstein Series. The Function ?.- 17. Modular Functions and Forms. The Modular Figure, Discontinuous Groups of Automorphisms.- 18. The Multiplicity Formula for Zeros of a Modular Form. Dimension of Vector Spaces M°(k, ?) of Cusp Forms.- 19. Mapping Properties of j. Picard Theorem. Elliptic Curves. Jacobi’s Inversion Problem. Abel’s Theorem.- 20. Uniformization Principle. Automorphic Forms. Riemann—Roch Theorem and Its Consequences. Historical Sketch.- 21. Appendices. Exercises (Proofs of Theorems of Runge, Florack, Koebe, and Hurwitz. Triangle Groups. Elliptic Integrals and Transcendental Numbers).- XVII. Normal and Paracompact Spaces. Partition of Unity..- 1. Locally Compact Spaces Countable at Infinity.- 2. Normal Spaces. Urysohn’s Lemma.- 3. Extendibility of Continuous Functions on Normal Spaces.- 4. Tychonoff Spaces. Uniformizability. Compactification.- 5. The Theory of Maximal Ideals.- 6. The Gel’fand Theory of Maximal Ideals.- 7. Connection with Quantum Mechanics.- 8. Locally Finite Families.- 9. Paracompact Spaces. Partition of Unity. Metric Spaces are Paracompact.- XVIII. Measurable Mappings. The Transport of a Measure. Convolutions of Measures and Functions.- 1. Measurable Mappings.- 2. Topologies Determined by Families of Mappings.- 3. The Transport of a Measure.- 4. The Projective Limits of Hausdorff Spaces. Infinite Tensor Products and the Projective Limits of Measures.- 5. Convolutions of Measures and Functions.- 6. Convolutions of Functions and Measures on Rp.- 7. Convolutions of Integrable Functions.- XIX. The Theory of Distributions. Harmonic Analysis.- 1. The Space C0? (?).- 2. A Differentiate Partition of Unity on Rn.- 3. The Space of Test Functions. Distributions.- 4. Inductive Limits. The Topology of the Space D.- 5. The Pasting Together Principle for Distributions. The Support of a Distribution.- 6. The Space e(?). Distributions with Compact Supports.- 7. Operations on Distributions.- 8. The Convolution Algebra e (Rn).- 9. The Image of a Distribution.- 10. Remarks on the Tensor Products E??F and E??F. The Kernel Theorem.- 11. The Tensor Product E ? ?F of Hilbert Spaces.- 12. Regularization of Distributions.- 13. Examples of Distributions Important for Applications.- 14. The Fourier Transformation. The Space l.- 15. The Fourier Transformation as a Unitary Operator on the Space l2(Rn).- 16. Tempered Distributions. The Fourier Transformation in l’.- 17. The Laplace-Fourier Transformation for Functions and Distributions. The Paley-Wiener-Schwartz Theorem.- 18. Fundamental Solutions of Differential Operators.- 19. Positive-Definite Functions. Positive Distributions. The Theorems of Bochner and Minlos.- 20. Representations of Locally Compact Groups. The Relation between Unitary Representations and Positive-Definite Functions.- 21. The Haar Integral.- Index of Symbols.- Name Index.
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