Treatise on Analysis. Foundations of modern analysis

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