دانلود کتاب Treatise on Analysis I-VIII (8-Vols all in one)

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I (2ed. Enl. and Corr. printing, 1969)
Preface to the Enlarged and Corrected Printing
Preface
Contents
Notations
1. ELEMENTS OF THE THEORY OF SETS
1. Elements and sets
2. Boolean algebra
3. Product of two sets
4. Mappings
5. Direct and inverse images
6. Surjective, injective, and bijective mappings
7. Composition of mappings
8. Families of elements. Union, intersection, and products of families of sets. Equivalence relations
9. Denumerable sets
2. REAL NUMBERS
1. Axioms of the real numbers
2. Order propertics of the real numbers
3. Least upper bound and greatest lower bound
3. METRIC SPACES
1. Distances and metric spaces
2. Examples of distances
3. Isometries
4. Balls, spheres, diameter
5. Open sets
6. Neighborhoods
7. Interior of a set
8. Closed sets, cluster points, closure of a set
9. Dense subsets; separable spaces
10. Subspaces of a metric space
11. Continuous mappings
12. Homeomorphisms. Equivalent distances
13. Limits
14. Cauchy sequences, complete spaces
15. Elementary extension theorems
16. Compact spaces
17. Compact sets
18. Locally compact spaces
19. Connected spaces and connected sets
20. Product of two metric spaces
4. ADDITIONAL PROPERTIES OF THE REAL LINE
1. Continuity of algebraic operations
2. Monotone functions
3. Logarithms and exponentials
4. Complex numbers
5. The Tietze-Urysohn extension theorem
5. NORMED SPACES
1. Normed spaces and Banach spaces
2. Series in a normed space
3. Absolutely convergent series
4. Subspaces and finite products of normed spaces
5. Condition of continuity of a multilinear mapping
6. Equivalent norms
7. Spaces of continuous multilinear mappings
8. Closed hyperplanes and continuous linear forms
9. Finite dimensional normed spaces
10. Separable normed spaces
6. HILBERT SPACES
1. Hermitian forms
2. Positive hermitian forms
3. Orthogonal projection on a complete subspace
4. Hilbert sum of Hilbert spaces
5. Orthonormal systems
6. Orthonormalization
7. SPACES OF CONTINUOUS FUNCTIONS
1. Spaces of bounded functions
2. Spaces of bounded continuous functions
3. The Stone-Weierstrass approximation theorem
4. Applications
5. Equicontinuous sets
6. Regulated functions
8. DIFFERENTIAL CALCULUS
1. Derivative of a continuous mapping
2. Formal rules of derivation
3. Derivatives in spaces of continuous linear functions
4. Derivatives of functions of one variable
5. The mean value theorem
6. Applications of the mean value theorem
7. Primitives and integrals
8. Application: the number e
9. Partial derivatives
10. Jacobians
11. Derivative of an integral depending on a parameter
12. Higher derivatives
13. Differential operators
14. Taylor’s formula
9. ANALYTIC FUNCTIONS
1. Power series
2. Substitution of power series in a power series
3. Analytic functions
4. The principle of analytic continuation
5. Examples of analytic functions; the exponential function; the number π
6. Integration along a road
7. Primitive of an analytic function in a simply connected domain
8. Index of a point with respect to a circuit
9. The Cauchy formula
10. Characterization of analytic functions of complex variables
11. Liouville’s theorem
12. Convergent sequences of analytic functions
13. Equicontinuous sets of analytic functions
14. The Laurent series
15. Isolated singular points; poles; zeros; residues
16. The theorem of residues
17. Meromorphic functions
Appendix 9. APPLICATION OF ANALYTIC FUNCTIONS TO PLANE TOPOLOGY (Eilenberg\'s Method)
1. Index of a point with respect to a loop
2. Essential mappings in the unit circle
3. Cuts of the plane
4. Simple arcs and simple closed curves
10. EXISTENCE THEOREMS
1. The method of successive approximations
2. Implicit functions
3. The rank theorem
4. Differential equations
5. Comparison of solutions of differential equations
6. Linear differential equations
7. Dependence of the solution on parameters
8. Dependence of the solution on initial conditions
9. The theorem of Frobenius
11. ELEMENTARY SPECTRAL THEORY
1. Spectrum of a continuous operator
2. Compact operators
3. The theory of F. Riesz
4. Spectrum of a compact operator
5. Compact operators in Hilbert spaces
6. The Fredholm integral equation
7. The Sturm-Liouville problem
Apdx. ELEMENTS OF LINEAR ALGEBRA
1. Vector spaces
2. Linear mappings
3. Direct sums of subspaces
4. Bases. Dimension and codimension
5. Matrices
6. Multilinear mappings. Determinants
7. Minors of a determinant
References
Index
II (1ed. Enl. and Corr. printing, 1976)
Schematic Plan of the Work
Contents
Notation
12. TOPOLOGY AND TOPOLOGICAL ALGEBRA
1. Topological spaces
2. Topological concepts
3. Hausdorff spaces
4. Uniformizable spaces
5 . Products of uniformizable spaces
6. Locally finite coverings and partitions of unity
7. Semicontinuous functions
8. Topological groups
9. Metrizable groups
10. Spaces with operators. Orbit spaces
11. Homogeneous spaces
12. Quotient groups
13. Topological vector spaces
14. Locally convex spaces
15. Weak topologies
16. Baire\'s theorem and its consequences
13. INTEGRATION
1. Definition of a measure
2. Real measures
3. Positive measures. The absolute value of a measure
4. The vague topology
5. Upper and lower integrals with respect to a positive measure
6. Negligible functions and sets
7. Integrable functions and sets
8. Lebesgue\'s convergence theorems
9. Measurable functions
10. Integrals of vector-valued functions
11. The spaces L^1 and L^2
12. The space L^∞
13. Measures with base μ
14. Integration with respect to a positive measure with base μ
15. The Lebesgue-Nikodym theorem and the order relation on M_R(X)
16. Applications: I. Integration with respect to a complex measure
17. Applications: II. Dual of L^1
18. Canonical decompositions of a measure
19. Support of a measure. Measures with compact support
20. Bounded measures
21. Product of measures
14. INTEGRATION IN LOCALLY COMPACT GROUPS
1. Existence and uniqueness of Haar measure
2. Particular cases and examples
3. The modulus function on a group. The modulus of an automorphism
4. Haar measure on a quotient group
5. Convolution of measures on a locally compact group
6. Examples and particular cases of convolution of measures
7. Algebraic properties of convolution
8. Convolution of a measure and a function
9. Examples of convolutions of measures and functions
10. Convolution of two functions
11. Regularization
15. NORMED ALGEBRAS AND SPECTRAL THEORY
1. Normed algebras
2. Spectrum of an element of a normed algebra
3. Characters and spectrum of a commutative Banach algebra. The Gelfand transformation
4. Banach algebras with involution. Star algebras
5. Representations of algebras with involution
6. Positive linear forms, positive Hilbert forms, and representations
7. Traces, bitraces, and Hilbert algebras
8. Complete Hilbert algebras
9. The Plancherel-Godement theorem
10. Representations of algebras of continuous functions
11. The spectral theory of Hilbert
12. Unbounded normal operators
13. Extensions of hermitian operators
References
Volume II
Index
ERRATUM to Volume II, p.296
III (1ed., 1972)
Schematic Plan of the Work
Contents
Notation
16. DIFFERENTIAL MANIFOLDS
1. Charts, atlases, manifolds
2. Examples of differential manifolds. Diffeomorphisms
3. Differentiable mappings
4. Differentiable partitions of unity
5. Tangent spaces, tangent linear mappings, rank
6. Products of manifolds
7. Immersions, submersions, subimmersions
8. Submanifolds
9. Lie groups
10. Orbit spaces and homogeneous spaces
11. Examples: unitary groups, Stiefel manifolds, Grassmannians, projective spaces
12. Fibrations
13. Definition of fibrations by means of charts
14. Principal fiber bundles
15. Vector bundles
16. Operations on vector bundles
17. Exact sequences, subbundles, and quotient bundles
18. Canonical morphisms of vector bundles
19. Inverse image of a vector bundle
20. Differential forms
21. Orientable manifolds and orientations
22. Change of variables in multiple integrals. Lebesgue measures
23. Sard\'s theorem
24. Integral of a differential n-form over an oriented pure manifold of dimension n
25. Embedding and approximation theorems. Tubular neighborhoods
26. Differentiable homotopies and isotopies
27. The fundamental group of a connected manifold
28. Covering spaces and the fundamental group
29. The universal covering of a differential manifold
30. Covering spaces of a Lie group
17. DIFFERENTIAL CALCULUS ON A DIFFERENTIAL MANIFOLD I. Distributions and Differential Operators
1. The spaces E^{(r)} (U) (U open in R^n)
2. Spaces of C^∞ (resp. C\') sections of vector bundles
3. Currents and distributions
4. Local definition of a current. Supportof a current
5. Currents on an oriented manifold. Distributions on R^n
6. Real distributions. Positive distributions
7. Distributions with compact support. Point-distributions
8. The weak topology on spaces of distributions
9. Example: finite parts of divergent integrals
10. Tensor products of distributions
11. Convolution of distributions on a Lie group
12. Regularization of distributions
13. Differential operators and fields of point-distributions
14. Vector fields as differential operators
15. The exterior differential of a differential p-form
16. Connections in a vector bundle
17. Differential operators associated with a connection
18. Connections on a differential manifold
19. The covariant exterior differential
20. Curvature and torsion of a connection
Apdx. MULTILINEAR ALGEBRA
8. Modules. Free modules
9. Duality for free modules
10. Tensor product of free modules
11. Tensors
12. Symmetric and antisymmetric tensors
13. The exterior algebra
14. Duality in the exterior algebra
15. Interior products
16. Nondegenerate alternating bilinear forms. Symplectic groups
17. The symmetric algebra
18. Derivations and antiderivations of graded algebras
19. Lie algebras
References
Volume III
Index
IV (1ed., 1974)
Schematic Plan of the Work
Contents
Notation
18. DIFFERENTIAL CALCULUS ON A DIFFERENTIAL MANIFOLD II. Elementary Global Theory of 1st- and 2nd- Order Differential Equations. Elementary Local Theory of Differential Systems
1. First-order differential equations on a differential manifold
2. Flow of a vector field
3. 2nd-order differential equations on a manifold
4. Sprays and isochronous 2nd-order equations
5. Convexity properties of isochronous differential equations
6. Geodesics of a connection
7. One-parameter families of geodesics and Jacobi fields
8. Fields of p-directions, Pfaffian systems, and systems of partial differential equations
9. Differential systems
10. Integral elements of a differential system
11. Formulation of the problem of integration
12. The Cauchy-Kowalewska theorem
13. The Cartan-Kähler theorem
14. Completely integrable Pfaffian systems
15. Singular integral manifolds; characteristic manifolds
16. Cauchy characteristics
17. Examples: I. 1st-order partial differential equations
18. Examples: II. 2nd-order partial differential equations
19. LIE GROUPS AND LIE ALGEBRAS
1. Equivariant actions of Lie groups on fiber bundles
2. Actions of a Lie group G on bundles over G
3. The infinitesimal algebra and the Lie algebra of a Lie group
4. Examples
5. Taylor’s formula in a Lie group
6. The enveloping algebra of the Lie algebra of a Lie group
7. Immersed Lie groups and Lie subalgebras
8. Invariant connections, one-parameter subgroups, and the exponential mapping
9. Properties of the exponential mapping
10. Closed subgroups of real Lie groups
11. The adjoint representation. Normalizers and centralizers
12. The Lie algebra of the commutator group
13. Automorphism groups of Lie groups
14. Semidirect products of Lie groups
15. Differential of a mapping into a Lie group
16. Invariant differential forms and Haar measure on a Lie group
17. Complex Lie groups
20. PRINCIPAL CONNECTIONS AND RIEMANNIAN GEOMETRY
1. The bundle of frames of a vector bundle
2. Principal connections on principal bundles
3. Covariant exterior differentiation attached to a principal connection. Curvature form of a principal connection
4. Examples of principal connections
5. Linear connections associated with a principal connection
6. The method of moving frames
7. G-structures
8. Generalities on pseudo-Riemannian manifolds
9. The Levi-Civita connection
10. The Riemann-Christoffel tensor
11. Examples of Riemannian and pseudo-Riemannian manifolds
12. Riemannian structure induced on a submanifold
13. Curves in Riemannian manifolds
14. Hypersurfaces in Riemannian manifolds
15. The immersion problem
16. The metric space structure of a Riemannian manifold: local properties
17. Strictly geodesically convex balls
18. The metric space structure of a Riemannian manifold: global properties. Complete Riemannian manifolds
19. Periodic geodesics
20. 1st and 2nd variation of arclength. Jacobi fields on a Riemannian manifold
21. Sectional curvature
22. Manifolds with positive sectional curvature or negative sectional curvature
23. Riemannian manifolds of constant curvature
Apdx. TENSOR PRODUCTS AND FORMAL POWER SERIES
20. Tensor products of infinite-dimensional vector spaces
21. Algebras of formal power series
References
VOLUME IV
Index
V (1ed., 1977)
Contents
Schematic Plan of the Work
Notation
21. COMPACT LIE GROUPS AND SEMISIMPLE LIE GROUPS
1. Continuous unitary representations of locally compact groups
2. The Hilbert Algebra of a compact group
3. Characters of a compact group
4. Continuous unitary representations of compact groups
5. Invariant bilinear forms; the Killing form
6. Semisimple Lie groups. Criterion of semisimplicity for a compact Lie group
7. Maximal tori in compact connected Lie groups
8. Roots and almost simple subgroups of rank 1
9. Linear representations of SU(2)
10. Properties of the roots of a compact semisimple group
11. Bases of a root system
12. Examples: the classical compact groups
13. Linear representations of compact connected Lie groups
14. Anti-invariant elements
15. Weyl\'s formulas
16. Center, fundamental group and irreducible representations of semisimple compact connected groups
17. Complexifications of compact connected semisimple groups
18. Real forms of the complexifications of compact connected semisimple groups and symmetric spaces
19. Roots of a complex semisimple Lie algebra
20. Weyl bases
21. The Iwasawa decomposition
22. Cartan\'s criterion for solvable Lie algebras
23. E. E. Levi\'s theorem
Apdx. MODULES
22. Simple modules
23. Semisimple modules
24. Examples
25. The canonical decomposition of an endomorphism
26. Finitely generated Z-modules
References
V and VI
Index
VI (1ed., 1978)
Contents
Schematic Plan of the Work
Notation
22. HARMONIC ANALYSIS
1. Continuous functions of positive type
2. Measures of positive type
3. Induced representations
4. Induced representations and restrictions of representations to subgroups
5. Partial traces and induced representations of compact groups
Banach space
6. Gelfand pairs and spherical functions
7. Plancherel and Fourier transforms
8. The spaces P(G) and P\'(Z)
9. Spherical functions of positive type and irreducible representations
10. Commutative harmonic analysis and Pontrjagin duality
11. Dual of a subgroup and of a quotient group
12. Poisson\'s formula
13. Dual of aproduct
14. Examples of duality
15. Continuous unitary representations of locally compact commutative groups
16. Declining functions on R^n
17. Tempered Distributions
18. Convolution of tempered distributions and the Paley-Wiener theorem
19. Periodic distributions and Fourier series
20. Sobolev spaces
References
Volume V and VI
Index
17. Tempered distributions
VII (1ed., 1988)
Notation
23. LINEAR FUNCTIONAL EQUATIONS. Part I. Pseudodifferential Operators
Part I. Pseudodifferential Operators
1. Integral Operators
2. Integral Operators of Proper Type
3. Integral Operators on Vector Bundles
4. Density Bundle and Kernel Sections
5. Bounded Sections
6. Volterra Operators
7. Carleman Operators
8. Generalized Eigenfunctions
9. Kernel Distributions
10. Regular Kernel Distributions
11. Smoothing Operators and Composition of Operators
12. Wave Front of a Distribution
13. Convolution Equations
14. Elementary Solutions
15. Problems of Existence and Uniqueness for Systems of Linear Partial Differential Equations
16. Operator Symbols
17. Oscillating Integrals
18. Lax-Maslov Operators
19. Pseudo-Differential Operators
20. Symbol of a Pseudodifferential Operator of Proper Type
21. Matrix Pseudodifferential Operators
22. Parametrix of Elliptical Operators on an Open Subset of R^n
23. Pseudodifferential Operators in H^s_0(X) Spaces
24. Classical Dirichlet Problem and Coarse Dirichlet Problems
25. The Green Operator
26. Pseudodifferential Operators on a Manifold
27. Adjoint of a Pseudodifferential Operator on a Manifold. Composition of Two Pseudodifferential Operators on a Manifold
28. Extension of Pseudodifferential Operators to Distribution Sections
29. Principal Symbols
30. Parametrix of Elliptic Operators on Manifolds
31. Spectral Theory of Hermitian Elliptic Operators: I. Self-Adjoint Extensions and Boundary Conditions
32. Spectral Theory of Hermitian Elliptic Operators: II. Generalized Eigenfunctions
33. Essentially Self-Adjoint Pseudodifferential Operators: I. Hermitian Convolution Operators on R^n
34. Essentially Self-Adjoint Pseudodifferential Operators: II. Atomic Spectra
35. Essentially Self-Adjoint Pseudodifferential Operators: III. Hermitian Elliptic Operators on a Compact Manifold
36. Invariant Differential Operators
37. Differential Properties of Spherical Functions
38. Example: Spherical Harmonics
References
VII and VIII
Index
VIII (1ed., 1993)
Contents
Notation
23. LINEAR FUNCTIONAL EQUATIONS. Part II. Boundary Value Problems
Part II. Boundary Value Problems
39. Weyl-Kodaira theory : I. Elliptic differential operators on an interval of R
40. Weyl-Kodaira theory : II. Boundary conditions
41. Weyl-Kodaira theory : III. Self-adjoint operators associated with a linear differential equation
42. Weyl-KodairaTheory : IV. Green Function and Spectrum
43. Weyl-Kodaira theory : V. The case of second order equations
44. Weyl-Kodaira theory : VI. Example : Second order equations with periodic coefficients
45. Weyl-Kodaira theory : VII. Example: Gelfand-Levitan equations
46. Multilayer potentials : I. Symbols of rational type
47. Multilayer potentials : II. The case of hyperplane multilayers
48. Multilayer potentials : III. General case
49. Fine boundary value problems for elliptic differential operators : I. The Calderon operator
50. Fine boundary value problems for elliptic differential operators : II. Elliptic boundary value problems
51. Fine boundary value problems for elliptic differential operators : III. Ellipticity criteria
52. Fine boundary value problems for elliptic differential operators : IV. The spaces H^{s, r}(U_+)
53. Fine boundary value problems for elliptic differential operators : V. H^{s, r}-spaces and P-potentials
54. Fine boundary value problems for elliptic differential operators : VI. Regularity on the boundary
55. Fine boundary value problems for elliptic differential operators : VII. Coercive problems
56. Fine boundary value problems for elliptic differential operators : VIII. Generalized Green\'s formula
57. Fine boundary value problems for elliptic differential operators : IX. Fine problems associated with coercive problems
58. Fine boundary value problems for elliptic differential operators : X. Examples
59. Fine boundary value problems for elliptic differential operators : XI. Extension to some non-hermitian operators
60. Fine boundary value problems for elliptic differential operators : XII. Case of second-order operators; Neumann\'s problem
61. Fine boundary value problems for elliptic differential operators : XIII. The maximum principle
62. Parabolic equations : I. Construction of a one-sided local resolvent
63. Parabolic equations : II. The one-sided global Cauchy problem
64. Parabolic equations : III. Traces and eigenvalues
65. Evolution distributions
66. The wave equation : I. Generalized Cauchy problem
67. The wave equation : II. Propagation and domain of influence
68. The wave equation : III. Signals, waves, and rays
69. Strictly hyperbolic equations : I. Preliminary results
70. Strictly hyperbolic equations : II. Construction of a local approximate resolvent
71. Strictly hyperbolic equations : III. Examples and variations
72. Strictly hyperbolic equations : IV. The Cauchy problem for strictly hyperbolic differential operators; existence and local uniqueness
73. Strictly hyperbolic equations : V. Global problems
74. Strictly hyperbolic equations : VI. Extension to manifolds
75. Application to the spectrum of a hermitian elliptic operator
References
VII and VIII
Index