Irina Meghea

Ekeland variational principle
with
generalizations
and
variants















CONTENTS

Introduction 1
Chapter I. Ekeland variational principle 3
1 Ekeland priciple in complete metric spaces 5
Several applications of Ekeland principle 30
2 Ekeland principle versus equivalent theorems 63
A. Ekeland principle versus the Drop theorem and the Flower
Petal theorem 63
Support functional and support point for closed convex sets 71
The surjectivity of nonlinear operators 76
B. Ekeland principle versus Caristi - Kirk theorem 81
C. Ekeland principle versus Takahashi theorem, Caristi - Kirk
theorem and Oettli - Théra theorem 83
D. Siegel theorem 88
3 Ekeland principle in Banach spaces 95
 - derivative 105
4 Palais - Smale type conditions 113
(PS)c 114
(PS)c, F 117
Critical points and weak solutions for an elliptic type equation 120
5 Subdifferential calculus 125
A. Subderivative 125
B. Superderivative 141
C. Directional derivative 141
D.  - subderivative and  - superderivative 147
E. The property (H) and the Banach space D (X) 149
F. Clarke derivative 153
G. Clarke subderivative 156
Clarke subderivative 156
Strict derivative of Hadamard type 162
Rules of subdifferential calculus 166
H. Critical points for nondifferentiable functionals 178
I. Mathematical programming 185
6 Variational theorems of minimax type in Banach spaces 187
Minimax theorem I 187
Minimax theorem II 197
Mountain pass theorem and Saddle point theorem for locally
Lipschitz functions 211
Minimax theorem III 218
7 Variational theorems of minimax type on Finsler manifolds 225
Minimax theorem I 229
Minimax theorem II (strong form) 234
Ljusternik - Schnirelmann theorem 245
Minimax theorem III (relaxed boundary conditions) 249
Relaxed Mountain pass and Saddle point theorems 255
Critical orbits of the real periodic functionals 259
8 Ekeland principle for vector - valued functions and minimal
points in product spaces 271
Preliminaries of affine geometry 271
Minimal point theorem I 275
Minimal point theorem II 279
Minimal point theorem III 281
Minimal point theorem IV 288
9 Other variants of Ekeland principle 295
A. Vectorial variant of Ekeland principle 295
B. Vectorial variant of Ekeland principle 305
C. Vectorial variant of Ekeland principle 310
D. Vectorial generalization of the Ekeland and Caristi - Kirk
theorems 314
E. Multivalued variant of Ekeland principle 321
F. Multivalued variants of Ekeland principle 328
G. Ekeland principle in sequentially complete locally convex
spaces 339
H. Ekeland principle in locally complete locally convex spaces 353
Local completeness 354
Two l-complete variants of Ekeland principle 355
The l-complete versions of Ekeland principle face to the l-complete
version of Daneš drop theorem 358
I. Ekeland principle in uniform spaces 361
Introduction 362
Notations 362
Uniform space 363
The topology of a uniform space 364
Pseudo-distance 368
Family of quasi-distances 369
Ekeland principle in uniform spaces (families of quasi-distances) 371
Ekeland principle in uniform spaces (generalized families of quasi-
distances) 377
Generalized family of quasi-distances 377
Minimal point theorem 378
J. Ekeland principle face to the w-distance 382
10. Ekeland principle and Caristi - Kirk theorem in
probabilistic metric spaces 391
A. Menger PM-space 391
Distribution function 391
t-norm 392
Menger PM-space 392
B. Caristi - Kirk theorem in probabilistic metric spaces 396
C. Ekeland principle in probabilistic metric spaces 398

Chapter II. Smooth perturbed variational principles 403
1. Borwein - Preiss principle 405
2. Deville - Godefroy - Zizler principle 417
3. Ghoussoub - Maurey theorems 427
4. Generalization and unification of the Ekeland and Borwein -
Preiss principles 445
Annex 459
Differential manifolds 459
Finsler manifolds 477
Riemann manifolds 482
Bibliography 487
Additional bibliography (with brief presentations) 495
Index 521




















INTRODUCTION





In the frame of Variational Calculus, the elementary proposition
“If  : X  R , X real normed space,  has in x0 a local minimum point (hence in particular a global minimum point) and it is Gâteaux differentiable at x0 , then (x 0) = 0” (x0 is critical point)
is called variational principle ([30], 1; w comes from weak).
This is the reason why the neighbour propositions, for instance those in which X is replaced by a metric space, or in which the statement “(x0) = 0” is replaced by “,  > 0 any“ etc, are called variational principles (perturbed). The adjective “perturbed” is imposed by the fact that not the function  is minimized, but a function of the form  +  (Ekeland), or of more general form  + (Borwein - Preiss), or of even more general form  + f (Deville- Godefroy - Zizler), f having some given properties (the second term is the perturbation function).
The Ekeland principle is a perturbed variational principle discovered in 1972 ([30], 1) and nowadays, after more than 30 years, it was proved to be, and this monograph will strongly support this, the foundation of the modern Variational Calculus (see, for instance, the minimax theorems in Banach spaces or in the Finsler manifolds, in which the key step of demonstration is made by the application of the Ekeland principle).
As referring the applications, these are numerous and various: the geometry of Banach spaces, nonlinear analysis, differential equations and partial differential equations, global analysis, probabilistic analysis, differential geometry, fixed point theorems, nonlinear semigroups, dynamical systems, optimization, mathematical programming, optimal control.
Finally, we cannot close this INTRODUCTION without the confession of Ekeland ([30], 4):
“The grandfather of these all is the celebrated 1961 theorem of Bishop and Phelps that the set of continuous linear functionals on a Banach space E, which attain their maximum on a prescribed non void closed convex bounded subset X  E, is norm-dense in E*”.
This Bishop-Phelps theorem can be found in 2.35, Chapter I, 2.

*

* *
All the normed spaces that appear in this monograph are real.


Let f : X  ( , + ] be given. The domain of f , dom f , is
dom f {x X : f ( x ) < +}.
f is proper dom f  .
N will denote the set of integer numbers  1.
Both for the functions and for the sequences will be used the terms increasing and strictly increasing, decreasing and strictly decreasing instead of nondecreasing and increasing respectively nonincreasing and decreasing.




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DIFFERENTIAL CALCULUS
AND
INTEGRAL CALCULUS





A TREATISE FOR MATHEMATICIANS, ENGINEERS, PHYSICISTS AND CHEMISTS


In three volumes



This treatise in three massive volumes (vol. I, 840 pages, Ed. Tehnica, Bucharest, 1998; vol. II, 1000 pages, Ed. Tehnica, Bucharest, 2000; vol. III, 800 pages, Printech, Bucharest, 2002), encompasses a vast exposure of the differential and integral calculus in Rn and in C. It is based on Zermelo-Fraenkel axiom system and, except Chapter 0, it is self-contained: every notion used is defined and every statement used is demonstrated. About the entire treatise is performed a special and permanent supervision of the notations, statements and methods which have, or even might have, an impact on nature sciences and techniques. Moreover, implementation of topology, functional analysis and also of some more abstract theories (integration on differential manifolds in Rn, integration with respect to a measure on a  - algebra, etc.) is massive but supported on a hand by a torrent of 2000 solved exercises and also by figures, these clarifying the concepts or being an intuitive support and calming the abstract character, and on the other hand, by numerous applications for which functional analysis especially offers quite simple and powerful techniques.





CONTENTS
VOL. I
Differential Calculus

CHAPTER 0. Miscellanea (Latin)
CHAPTER I. Series of complex numbers
§1. Sequences of real numbers
1.Upper and lower limit
2.Sequences of complex numbers
3.Equations with finite differences and Tchebychef polynomials
4.Fibonacci numbers
5.Nonlinear equation with a real unknown
6.Order of convergence
§2. Series of complex numbers
1.Cauchy theorem
2.Rule of Abel
3.Absolute convergence
4.Comparison criteria
5.Criteria of Cauchy and D’Alembert
6.Criteria of Kummer, Raabe-Duhamel, Bertrand, Gauss
7.Taylor formula and Taylor series
8.The formulae of Wallis and Stirling
9.Finite differences, quotient differences, Newton interpolation formulae, Gauss interpolation formulae, interpolation Hermite polynomial
10.Calculus with approximation of the sum and the speeding of the convergence
11.Summation of the divergent series, Césaro and Abel-Poisson summations
12.Continued fraction
13.Calculus of the elementary function values
14.Double series
15.Infinite product

CHAPTER II. Elements of general topology
§1 First notions
1.Metric space
2.Topological space
3.Remarkable points
§2. Limit and continuity
1.Limit
2.Continuity
3.Semicontinuity
§3. Complete metric space
1.Cauchy sequence, complete metric space
2.Fixed point theorems
3.Nonlinear equation with a real unknown (Methods of Tchebychef, Steffensen, Schröder)


§4. Compact, locally compact and paracompact spaces
§5. Connected and locally connected spaces

CHAPTER III. Elements of functional analysis
§1. Norm and topology
1.Normed space
2.Topology of the normed space
3.Series in normed space
4.Prehilbert space
5.Normed algebra
§2. Continuous multilinear map
§3. Normed spaces with finite
dimension
1.Tychonov theorem
2.Riesz theorem
§4. Matrix analysis
1.Eigenvalues and eigenvectors
2.Hermite matrices
3.Minimal polynomials and functions of matrices
4.Norms of matrices
5.Iteration methods for linear equations systems (Seidel method, surrelaxation method, Tchebychef-Richardson method)
6.Supplements for eigenvalues (theorems of Perron - Frobenius and Hurwitz)
§5. Prehilbert space
Theorems of Schmidt, Riesz and Lax- Milgram, Gram- Schmidt lemma


§6. Sequences of linear operators
1.Banach-Steinhaus theorem
2.Summation of divergent series
3.Approximation of definite integrals (formulae of Newton-Côtes, Gauss, Tchebychef)

CHAPTER IV. Differential calculus in Rn and in normed spaces
§1. Partial derivative and Hölder continuity, function with bounded variation, Schwarz theorem completions
§2. Differential
1.Derivatives of Fréchet and Gâteaux
2.Differential of functions with values in Rm
3.Fréchet derivative of a composite function
4.Formula of Taylor-Lagrange-Cauchy
5.Differentials of higher order
6.Approximate derivative
§3. Local extrema
1.Bilinear forms
2.Quadratic forms on vector spaces, Gram determinant
3.Inner local extrema
4.Global extrema
§4. Implicit functions
1.Implicit function
2.System of implicit functions
3.Functional dependence
§5. Elements of geometry
1.Multilinear alternated form
2.Euclidian vector space
3.Analytic geometry (A. Affine geometry, B. Euclidean geometry)
4.Differential geometry in Euclidean space R3
A. Parameterized curve,
B. Smooth curve,
C. Smooth surface (first
fundamental form, isometry,
orientation, the Gauss indicator,
second fundamental form, normal
curvature, principal curvatures, curvature line, gaussian curvature, middle curvature, asymptotic line, surface removals, the formulae of Gauss, Weingarten, Peterson-Codazzi, geodesic, smooth surface with constant gaussian curvature, Riemann surface in the Euclidean space R3)
5.Parametrized curve in
Euclidean space Rn
6.Differential manifold in Rn
7.Conditional local extrema


§6. Differential calculus in normed spaces
1. Fréchet partial derivative
2. Implicit function
3. Inner local extrema and almost critical points
§7. Convex analysis in Rn and normed spaces
1.Convex set
2.Convex function
3.Conditions of Fritz John and Kuhn-Tucker
4.Simplex in Rn (theorems of Brouwer and Schauder- Tychonov)
§8. Nonlinear equations
1.Orders of convergence
2.Newton method in Rn
3.Other methods (nonlinear SOR, Newton-SOR, Broyden)
§9. Multivalued functions
1.Hausdorff distance, Vietoris topologies
2.Closed multivalued function
3.Fixed point theorems (Banach- Nadler, Kakutani, Kakutani-Glicksberg)

ANNEX I
Zermelo-Fraenkel axiom system
ANNEX II
Peano axiom system



VOL. II

INTEGRAL CALCULUS

CHAPTER V. Uniform limit.
Generalized Riemann integral, Cauchy integral
§1. Generalized Riemann integral
1.Riemann integral on unbounded interval
2.Riemann integral with singular point
3.Riemann integral on compact interval with parameter
4.Complex function with a real variable
§2. Generalized Riemann integral
with parameter
1.Uniform limit
2.Euler functions  and B
§3. Sequences and series of functions.
First elements of complex
analysis
1.Sequence of functions
2.Series of functions
3.First elements of complex analysis
4.Taylor series
5.Analytic real functions
6.Dirichlet series
7.Newton series

§4. Approximations of real and
complex functions
1.The Stone-Weierstrass theorem
2.Weierstrass theorem of approximation
3.Equicontinuity
4.Tchebychef approximation
5.Cubic spline approximation
§5. Other elements of complex analysis, Cauchy-Weierstrass theory and conformal mapping
1.Cauchy integral
2.Cauchy fundamental theorem (preliminaries versions)
3.Integral formula of Cauchy (preliminaries versions)
4.Analytic complex function
5.Theorems of Cauchy, Liouville, Schwarz, Hadamard
6.Poles and meromorphic functions
7.Laurent series
8.Local correspondence
9.Omological form of Cauchy fundamental theorem
10. The residue theorem
11. Calculus of definite integrals
12. The Riemann representation theorem, homotopic form of Cauchy theorem, theorems of Picard and Schottky
13. Runge theory of uniform approximation
14. Gamma function
15. Riemann functions and the theorem of Hadamard – de la Vallée Poussin
16. Asymptotic expansion
CHAPTER VI. Riemann integral in Rn
§1. Jordan measure in Rn
§2. Riemann integral in Rn
CHAPTER VII. Line integral in Rn and integral on smooth surface in R3
§1. Clairaut integral in Rn
1.Stieltjes integral
2.Line integral in Rn
3.Differential form of degree I
§2. Integration on smooth surface in R3

VOL. III

INTEGRAL CALCULUS

CHAPTER VIII. Lebesgue integral in Rn, the abstract integral and the integral on differential manifold in Rn
§1. Lebesgue measure in Rn and abstract
measure
1.Elementary sets in Rn
2.Lebesgue measure in Rn
3.Abstract measure
4.Axiom of choice and cardinal numbers
§2. Lebesgue measurable function and
measurable function
1.Lebesgue measurable function
2.Almost everywhere and almost uniform convergence
3.Convergence in measure
4.Measurable function


§3. Lebesgue integral in Rn and the integral
with respect to a measure on a σ-algebra
1.Lebesgue integrable complex and numerical function
2.Sequences of Lebesgue integrable functions
3.Numerical functions with Lebesgue integral
4.Change of variables in Lebesgue integral in Rn
5.Fubini theorem
6.Lebesgue integral in Rn with parameter
7.Integral with respect to a measure on a σ-algebra (inequalities of Hölder and Minkovski, theorems of Halmos, Vitali, Beppo-Levi, Fatou, Lebesgue, Fubini, Luzin, Radon-Nykodim, Vitali-Carathéodory and Hahn, Riesz representation theorem, Jensen inequality)
§4. The integral on differential manifold
in Rn
1.Measure and integral on smooth
manifolds in Rn
2.Orientation of differential manifolds in Rn
3.Differential forms of some degree (Elie Cartan exterior calculus)
4.Integration of differential forms on oriented smooth manifolds in Rn
5.Differential with border and pseudoborder manifold, orientation and integration
6.General formula Stokes - Ampère- Poincaré
7.Vector analysis
8.Harmonic functions in Rn


§5. The spaces Lp (X, μ)
1.Definitions and first properties
2.Density and separability
3.Uniform convexity
4.Duals
5.Convolution, regularization, relatively sets
6.Interpolation
7.Derivative of the ║∙║Lp norm


§6. Fourier series
1.Orthonormed basis in Hilbert spaces
2.Periodical function
3.Trigonometric series
4.Associated Fourier series
5.Féjer theorem (Dirichlet kernel, Féjer kernel)
6.Development in Fourier series and examples
7.Orthonormed basis in L2([a, b])
8.Other conditions of punctual and uniform convergence
9.Differentiation and integration of Fourier series
10.Theoretical aspects

CHAPTER IX. Complements of General Topology and Functional Analysis
§1 Complements of General Topology (comparison of topologies, initial and final topologies, Alexander theorem, product of topological spaces, product of Hausdorff spaces, the separation axioms (T1) – (T4), product of compact and locally compact spaces, compactifications, product of connected and locally connected spaces, axioms of countability, countably compact and sequentially compact spaces, metrisable spaces, separable spaces, Brower degree)
§2 Hahn - Banach theorem (real form, complex form, geometrical form, separation of convex sets, theorems of Dieudonné, Dunford-Schwartz, the Krein- Milman theorem, extension of linear operators)
§3 Weak topology and weak * topology (Mazur theorem, Alaoglu-Bourbaki-Kakutani theorem)
§4 Reflexive Spaces (theorems of Kakutani-Šmulian, Milman-Pettis- Kakutani, Eberlein - Šmulian , James, Krein- Šmulian, the best approximation, Day’s theorem)
§5 Topological vector spaces
1.Topological vector space
2.Locally convex space
3.Linear mappings
4.Duality
§6 Distributions
1.Test function spaces
2.Calculus with distributions
3.Localization
4.Supports of distributions
5.Convolutions
§7 Fourier transforms
1.Direct Fourier transform in L1(R)
2.Inverse Fourier transform in L1(R)
3.Direct Fourier transform in L2(R)
4.Tempered distributions
5.Paley-Wiener theorem
6.Sobolev lemma
7.Tauberian theory
§8. Laplace Transform
1.Laplace integral
2.Laplace transform
3.Laplace integral inversion
4.Representation by Laplace integral
5.Connection with Bessel functions
§9. Completions to Hilbert spaces (Elementary spectral theory)
1.Spectre of continuous operator
2.Compact operator
3.Riesz-Fredholm theory
4.Spectre of compact operator
5.Compact operators in Hilbert spaces
6.Spectral decomposition of compact self-adjoint operators

DIFFERENTIAL CALCULUS
AND
INTEGRAL CALCULUS





A TREATISE FOR MATHEMATICIANS, ENGINEERS, PHYSICISTS AND CHEMISTS


In three volumes



This treatise in three massive volumes (vol. I, 840 pages, Ed. Tehnica, Bucharest, 1998; vol. II, 1000 pages, Ed. Tehnica, Bucharest, 2000; vol. III, 800 pages, Printech, Bucharest, 2002), encompasses a vast exposure of the differential and integral calculus in Rn and in C. It is based on Zermelo-Fraenkel axiom system and, except Chapter 0, it is self-contained: every notion used is defined and every statement used is demonstrated. About the entire treatise is performed a special and permanent supervision of the notations, statements and methods which have, or even might have, an impact on nature sciences and techniques. Moreover, implementation of topology, functional analysis and also of some more abstract theories (integration on differential manifolds in Rn, integration with respect to a measure on a  - algebra, etc.) is massive but supported on a hand by a torrent of 2000 solved exercises and also by figures, these clarifying the concepts or being an intuitive support and calming the abstract character, and on the other hand, by numerous applications for which functional analysis especially offers quite simple and powerful techniques.





CONTENTS
VOL. I
Differential Calculus

CHAPTER 0. Miscellanea (Latin)
CHAPTER I. Series of complex numbers
§1. Sequences of real numbers
1.Upper and lower limit
2.Sequences of complex numbers
3.Equations with finite differences and Tchebychef polynomials
4.Fibonacci numbers
5.Nonlinear equation with a real unknown
6.Order of convergence
§2. Series of complex numbers
1.Cauchy theorem
2.Rule of Abel
3.Absolute convergence
4.Comparison criteria
5.Criteria of Cauchy and D’Alembert
6.Criteria of Kummer, Raabe-Duhamel, Bertrand, Gauss
7.Taylor formula and Taylor series
8.The formulae of Wallis and Stirling
9.Finite differences, quotient differences, Newton interpolation formulae, Gauss interpolation formulae, interpolation Hermite polynomial
10.Calculus with approximation of the sum and the speeding of the convergence
11.Summation of the divergent series, Césaro and Abel-Poisson summations
12.Continued fraction
13.Calculus of the elementary function values
14.Double series
15.Infinite product

CHAPTER II. Elements of general topology
§1 First notions
1.Metric space
2.Topological space
3.Remarkable points
§2. Limit and continuity
1.Limit
2.Continuity
3.Semicontinuity
§3. Complete metric space
1.Cauchy sequence, complete metric space
2.Fixed point theorems
3.Nonlinear equation with a real unknown (Methods of Tchebychef, Steffensen, Schröder)


§4. Compact, locally compact and paracompact spaces
§5. Connected and locally connected spaces

CHAPTER III. Elements of functional analysis
§1. Norm and topology
1.Normed space
2.Topology of the normed space
3.Series in normed space
4.Prehilbert space
5.Normed algebra
§2. Continuous multilinear map
§3. Normed spaces with finite
dimension
1.Tychonov theorem
2.Riesz theorem
§4. Matrix analysis
1.Eigenvalues and eigenvectors
2.Hermite matrices
3.Minimal polynomials and functions of matrices
4.Norms of matrices
5.Iteration methods for linear equations systems (Seidel method, surrelaxation method, Tchebychef-Richardson method)
6.Supplements for eigenvalues (theorems of Perron - Frobenius and Hurwitz)
§5. Prehilbert space
Theorems of Schmidt, Riesz and Lax- Milgram, Gram- Schmidt lemma


§6. Sequences of linear operators
1.Banach-Steinhaus theorem
2.Summation of divergent series
3.Approximation of definite integrals (formulae of Newton-Côtes, Gauss, Tchebychef)

CHAPTER IV. Differential calculus in Rn and in normed spaces
§1. Partial derivative and Hölder continuity, function with bounded variation, Schwarz theorem completions
§2. Differential
1.Derivatives of Fréchet and Gâteaux
2.Differential of functions with values in Rm
3.Fréchet derivative of a composite function
4.Formula of Taylor-Lagrange-Cauchy
5.Differentials of higher order
6.Approximate derivative
§3. Local extrema
1.Bilinear forms
2.Quadratic forms on vector spaces, Gram determinant
3.Inner local extrema
4.Global extrema
§4. Implicit functions
1.Implicit function
2.System of implicit functions
3.Functional dependence
§5. Elements of geometry
1.Multilinear alternated form
2.Euclidian vector space
3.Analytic geometry (A. Affine geometry, B. Euclidean geometry)
4.Differential geometry in Euclidean space R3
A. Parameterized curve,
B. Smooth curve,
C. Smooth surface (first
fundamental form, isometry,
orientation, the Gauss indicator,
second fundamental form, normal
curvature, principal curvatures, curvature line, gaussian curvature, middle curvature, asymptotic line, surface removals, the formulae of Gauss, Weingarten, Peterson-Codazzi, geodesic, smooth surface with constant gaussian curvature, Riemann surface in the Euclidean space R3)
5.Parametrized curve in
Euclidean space Rn
6.Differential manifold in Rn
7.Conditional local extrema


§6. Differential calculus in normed spaces
1. Fréchet partial derivative
2. Implicit function
3. Inner local extrema and almost critical points
§7. Convex analysis in Rn and normed spaces
1.Convex set
2.Convex function
3.Conditions of Fritz John and Kuhn-Tucker
4.Simplex in Rn (theorems of Brouwer and Schauder- Tychonov)
§8. Nonlinear equations
1.Orders of convergence
2.Newton method in Rn
3.Other methods (nonlinear SOR, Newton-SOR, Broyden)
§9. Multivalued functions
1.Hausdorff distance, Vietoris topologies
2.Closed multivalued function
3.Fixed point theorems (Banach- Nadler, Kakutani, Kakutani-Glicksberg)

ANNEX I
Zermelo-Fraenkel axiom system
ANNEX II
Peano axiom system



VOL. II

INTEGRAL CALCULUS

CHAPTER V. Uniform limit.
Generalized Riemann integral, Cauchy integral
§1. Generalized Riemann integral
1.Riemann integral on unbounded interval
2.Riemann integral with singular point
3.Riemann integral on compact interval with parameter
4.Complex function with a real variable
§2. Generalized Riemann integral
with parameter
1.Uniform limit
2.Euler functions  and B
§3. Sequences and series of functions.
First elements of complex
analysis
1.Sequence of functions
2.Series of functions
3.First elements of complex analysis
4.Taylor series
5.Analytic real functions
6.Dirichlet series
7.Newton series

§4. Approximations of real and
complex functions
1.The Stone-Weierstrass theorem
2.Weierstrass theorem of approximation
3.Equicontinuity
4.Tchebychef approximation
5.Cubic spline approximation
§5. Other elements of complex analysis, Cauchy-Weierstrass theory and conformal mapping
1.Cauchy integral
2.Cauchy fundamental theorem (preliminaries versions)
3.Integral formula of Cauchy (preliminaries versions)
4.Analytic complex function
5.Theorems of Cauchy, Liouville, Schwarz, Hadamard
6.Poles and meromorphic functions
7.Laurent series
8.Local correspondence
9.Omological form of Cauchy fundamental theorem
10. The residue theorem
11. Calculus of definite integrals
12. The Riemann representation theorem, homotopic form of Cauchy theorem, theorems of Picard and Schottky
13. Runge theory of uniform approximation
14. Gamma function
15. Riemann functions and the theorem of Hadamard – de la Vallée Poussin
16. Asymptotic expansion
CHAPTER VI. Riemann integral in Rn
§1. Jordan measure in Rn
§2. Riemann integral in Rn
CHAPTER VII. Line integral in Rn and integral on smooth surface in R3
§1. Clairaut integral in Rn
1.Stieltjes integral
2.Line integral in Rn
3.Differential form of degree I
§2. Integration on smooth surface in R3

VOL. III

INTEGRAL CALCULUS

CHAPTER VIII. Lebesgue integral in Rn, the abstract integral and the integral on differential manifold in Rn
§1. Lebesgue measure in Rn and abstract
measure
1.Elementary sets in Rn
2.Lebesgue measure in Rn
3.Abstract measure
4.Axiom of choice and cardinal numbers
§2. Lebesgue measurable function and
measurable function
1.Lebesgue measurable function
2.Almost everywhere and almost uniform convergence
3.Convergence in measure
4.Measurable function


§3. Lebesgue integral in Rn and the integral
with respect to a measure on a σ-algebra
1.Lebesgue integrable complex and numerical function
2.Sequences of Lebesgue integrable functions
3.Numerical functions with Lebesgue integral
4.Change of variables in Lebesgue integral in Rn
5.Fubini theorem
6.Lebesgue integral in Rn with parameter
7.Integral with respect to a measure on a σ-algebra (inequalities of Hölder and Minkovski, theorems of Halmos, Vitali, Beppo-Levi, Fatou, Lebesgue, Fubini, Luzin, Radon-Nykodim, Vitali-Carathéodory and Hahn, Riesz representation theorem, Jensen inequality)
§4. The integral on differential manifold
in Rn
1.Measure and integral on smooth
manifolds in Rn
2.Orientation of differential manifolds in Rn
3.Differential forms of some degree (Elie Cartan exterior calculus)
4.Integration of differential forms on oriented smooth manifolds in Rn
5.Differential with border and pseudoborder manifold, orientation and integration
6.General formula Stokes - Ampère- Poincaré
7.Vector analysis
8.Harmonic functions in Rn


§5. The spaces Lp (X, μ)
1.Definitions and first properties
2.Density and separability
3.Uniform convexity
4.Duals
5.Convolution, regularization, relatively sets
6.Interpolation
7.Derivative of the ║∙║Lp norm


§6. Fourier series
1.Orthonormed basis in Hilbert spaces
2.Periodical function
3.Trigonometric series
4.Associated Fourier series
5.Féjer theorem (Dirichlet kernel, Féjer kernel)
6.Development in Fourier series and examples
7.Orthonormed basis in L2([a, b])
8.Other conditions of punctual and uniform convergence
9.Differentiation and integration of Fourier series
10.Theoretical aspects

CHAPTER IX. Complements of General Topology and Functional Analysis
§1 Complements of General Topology (comparison of topologies, initial and final topologies, Alexander theorem, product of topological spaces, product of Hausdorff spaces, the separation axioms (T1) – (T4), product of compact and locally compact spaces, compactifications, product of connected and locally connected spaces, axioms of countability, countably compact and sequentially compact spaces, metrisable spaces, separable spaces, Brower degree)
§2 Hahn - Banach theorem (real form, complex form, geometrical form, separation of convex sets, theorems of Dieudonné, Dunford-Schwartz, the Krein- Milman theorem, extension of linear operators)
§3 Weak topology and weak * topology (Mazur theorem, Alaoglu-Bourbaki-Kakutani theorem)
§4 Reflexive Spaces (theorems of Kakutani-Šmulian, Milman-Pettis- Kakutani, Eberlein - Šmulian , James, Krein- Šmulian, the best approximation, Day’s theorem)
§5 Topological vector spaces
1.Topological vector space
2.Locally convex space
3.Linear mappings
4.Duality
§6 Distributions
1.Test function spaces
2.Calculus with distributions
3.Localization
4.Supports of distributions
5.Convolutions
§7 Fourier transforms
1.Direct Fourier transform in L1(R)
2.Inverse Fourier transform in L1(R)
3.Direct Fourier transform in L2(R)
4.Tempered distributions
5.Paley-Wiener theorem
6.Sobolev lemma
7.Tauberian theory
§8. Laplace Transform
1.Laplace integral
2.Laplace transform
3.Laplace integral inversion
4.Representation by Laplace integral
5.Connection with Bessel functions
§9. Completions to Hilbert spaces (Elementary spectral theory)
1.Spectre of continuous operator
2.Compact operator
3.Riesz-Fredholm theory
4.Spectre of compact operator
5.Compact operators in Hilbert spaces
6.Spectral decomposition of compact self-adjoint operators


Finis

DIFFERENTIAL CALCULUS
AND
INTEGRAL CALCULUS





A TREATISE FOR MATHEMATICIANS, ENGINEERS, PHYSICISTS AND CHEMISTS


In three volumes



This treatise in three massive volumes (vol. I, 840 pages, Ed. Tehnica, Bucharest, 1998; vol. II, 1000 pages, Ed. Tehnica, Bucharest, 2000; vol. III, 800 pages, Printech, Bucharest, 2002), encompasses a vast exposure of the differential and integral calculus in Rn and in C. It is based on Zermelo-Fraenkel axiom system and, except Chapter 0, it is self-contained: every notion used is defined and every statement used is demonstrated. About the entire treatise is performed a special and permanent supervision of the notations, statements and methods which have, or even might have, an impact on nature sciences and techniques. Moreover, implementation of topology, functional analysis and also of some more abstract theories (integration on differential manifolds in Rn, integration with respect to a measure on a  - algebra, etc.) is massive but supported on a hand by a torrent of 2000 solved exercises and also by figures, these clarifying the concepts or being an intuitive support and calming the abstract character, and on the other hand, by numerous applications for which functional analysis especially offers quite simple and powerful techniques.





CONTENTS
VOL. I
Differential Calculus

CHAPTER 0. Miscellanea (Latin)
CHAPTER I. Series of complex numbers
§1. Sequences of real numbers
1.Upper and lower limit
2.Sequences of complex numbers
3.Equations with finite differences and Tchebychef polynomials
4.Fibonacci numbers
5.Nonlinear equation with a real unknown
6.Order of convergence
§2. Series of complex numbers
1.Cauchy theorem
2.Rule of Abel
3.Absolute convergence
4.Comparison criteria
5.Criteria of Cauchy and D’Alembert
6.Criteria of Kummer, Raabe-Duhamel, Bertrand, Gauss
7.Taylor formula and Taylor series
8.The formulae of Wallis and Stirling
9.Finite differences, quotient differences, Newton interpolation formulae, Gauss interpolation formulae, interpolation Hermite polynomial
10.Calculus with approximation of the sum and the speeding of the convergence
11.Summation of the divergent series, Césaro and Abel-Poisson summations
12.Continued fraction
13.Calculus of the elementary function values
14.Double series
15.Infinite product

CHAPTER II. Elements of general topology
§1 First notions
1.Metric space
2.Topological space
3.Remarkable points
§2. Limit and continuity
1.Limit
2.Continuity
3.Semicontinuity
§3. Complete metric space
1.Cauchy sequence, complete metric space
2.Fixed point theorems
3.Nonlinear equation with a real unknown (Methods of Tchebychef, Steffensen, Schröder)


§4. Compact, locally compact and paracompact spaces
§5. Connected and locally connected spaces

CHAPTER III. Elements of functional analysis
§1. Norm and topology
1.Normed space
2.Topology of the normed space
3.Series in normed space
4.Prehilbert space
5.Normed algebra
§2. Continuous multilinear map
§3. Normed spaces with finite
dimension
1.Tychonov theorem
2.Riesz theorem
§4. Matrix analysis
1.Eigenvalues and eigenvectors
2.Hermite matrices
3.Minimal polynomials and functions of matrices
4.Norms of matrices
5.Iteration methods for linear equations systems (Seidel method, surrelaxation method, Tchebychef-Richardson method)
6.Supplements for eigenvalues (theorems of Perron - Frobenius and Hurwitz)
§5. Prehilbert space
Theorems of Schmidt, Riesz and Lax- Milgram, Gram- Schmidt lemma


§6. Sequences of linear operators
1.Banach-Steinhaus theorem
2.Summation of divergent series
3.Approximation of definite integrals (formulae of Newton-Côtes, Gauss, Tchebychef)

CHAPTER IV. Differential calculus in Rn and in normed spaces
§1. Partial derivative and Hölder continuity, function with bounded variation, Schwarz theorem completions
§2. Differential
1.Derivatives of Fréchet and Gâteaux
2.Differential of functions with values in Rm
3.Fréchet derivative of a composite function
4.Formula of Taylor-Lagrange-Cauchy
5.Differentials of higher order
6.Approximate derivative
§3. Local extrema
1.Bilinear forms
2.Quadratic forms on vector spaces, Gram determinant
3.Inner local extrema
4.Global extrema
§4. Implicit functions
1.Implicit function
2.System of implicit functions
3.Functional dependence
§5. Elements of geometry
1.Multilinear alternated form
2.Euclidian vector space
3.Analytic geometry (A. Affine geometry, B. Euclidean geometry)
4.Differential geometry in Euclidean space R3
A. Parameterized curve,
B. Smooth curve,
C. Smooth surface (first
fundamental form, isometry,
orientation, the Gauss indicator,
second fundamental form, normal
curvature, principal curvatures, curvature line, gaussian curvature, middle curvature, asymptotic line, surface removals, the formulae of Gauss, Weingarten, Peterson-Codazzi, geodesic, smooth surface with constant gaussian curvature, Riemann surface in the Euclidean space R3)
5.Parametrized curve in
Euclidean space Rn
6.Differential manifold in Rn
7.Conditional local extrema


§6. Differential calculus in normed spaces
1. Fréchet partial derivative
2. Implicit function
3. Inner local extrema and almost critical points
§7. Convex analysis in Rn and normed spaces
1.Convex set
2.Convex function
3.Conditions of Fritz John and Kuhn-Tucker
4.Simplex in Rn (theorems of Brouwer and Schauder- Tychonov)
§8. Nonlinear equations
1.Orders of convergence
2.Newton method in Rn
3.Other methods (nonlinear SOR, Newton-SOR, Broyden)
§9. Multivalued functions
1.Hausdorff distance, Vietoris topologies
2.Closed multivalued function
3.Fixed point theorems (Banach- Nadler, Kakutani, Kakutani-Glicksberg)

ANNEX I
Zermelo-Fraenkel axiom system
ANNEX II
Peano axiom system



VOL. II

INTEGRAL CALCULUS

CHAPTER V. Uniform limit.
Generalized Riemann integral, Cauchy integral
§1. Generalized Riemann integral
1.Riemann integral on unbounded interval
2.Riemann integral with singular point
3.Riemann integral on compact interval with parameter
4.Complex function with a real variable
§2. Generalized Riemann integral
with parameter
1.Uniform limit
2.Euler functions  and B
§3. Sequences and series of functions.
First elements of complex
analysis
1.Sequence of functions
2.Series of functions
3.First elements of complex analysis
4.Taylor series
5.Analytic real functions
6.Dirichlet series
7.Newton series

§4. Approximations of real and
complex functions
1.The Stone-Weierstrass theorem
2.Weierstrass theorem of approximation
3.Equicontinuity
4.Tchebychef approximation
5.Cubic spline approximation
§5. Other elements of complex analysis, Cauchy-Weierstrass theory and conformal mapping
1.Cauchy integral
2.Cauchy fundamental theorem (preliminaries versions)
3.Integral formula of Cauchy (preliminaries versions)
4.Analytic complex function
5.Theorems of Cauchy, Liouville, Schwarz, Hadamard
6.Poles and meromorphic functions
7.Laurent series
8.Local correspondence
9.Omological form of Cauchy fundamental theorem
10. The residue theorem
11. Calculus of definite integrals
12. The Riemann representation theorem, homotopic form of Cauchy theorem, theorems of Picard and Schottky
13. Runge theory of uniform approximation
14. Gamma function
15. Riemann functions and the theorem of Hadamard – de la Vallée Poussin
16. Asymptotic expansion
CHAPTER VI. Riemann integral in Rn
§1. Jordan measure in Rn
§2. Riemann integral in Rn
CHAPTER VII. Line integral in Rn and integral on smooth surface in R3
§1. Clairaut integral in Rn
1.Stieltjes integral
2.Line integral in Rn
3.Differential form of degree I
§2. Integration on smooth surface in R3

VOL. III

INTEGRAL CALCULUS

CHAPTER VIII. Lebesgue integral in Rn, the abstract integral and the integral on differential manifold in Rn
§1. Lebesgue measure in Rn and abstract
measure
1.Elementary sets in Rn
2.Lebesgue measure in Rn
3.Abstract measure
4.Axiom of choice and cardinal numbers
§2. Lebesgue measurable function and
measurable function
1.Lebesgue measurable function
2.Almost everywhere and almost uniform convergence
3.Convergence in measure
4.Measurable function


§3. Lebesgue integral in Rn and the integral
with respect to a measure on a σ-algebra
1.Lebesgue integrable complex and numerical function
2.Sequences of Lebesgue integrable functions
3.Numerical functions with Lebesgue integral
4.Change of variables in Lebesgue integral in Rn
5.Fubini theorem
6.Lebesgue integral in Rn with parameter
7.Integral with respect to a measure on a σ-algebra (inequalities of Hölder and Minkovski, theorems of Halmos, Vitali, Beppo-Levi, Fatou, Lebesgue, Fubini, Luzin, Radon-Nykodim, Vitali-Carathéodory and Hahn, Riesz representation theorem, Jensen inequality)
§4. The integral on differential manifold
in Rn
1.Measure and integral on smooth
manifolds in Rn
2.Orientation of differential manifolds in Rn
3.Differential forms of some degree (Elie Cartan exterior calculus)
4.Integration of differential forms on oriented smooth manifolds in Rn
5.Differential with border and pseudoborder manifold, orientation and integration
6.General formula Stokes - Ampère- Poincaré
7.Vector analysis
8.Harmonic functions in Rn


§5. The spaces Lp (X, μ)
1.Definitions and first properties
2.Density and separability
3.Uniform convexity
4.Duals
5.Convolution, regularization, relatively sets
6.Interpolation
7.Derivative of the ║∙║Lp norm


§6. Fourier series
1.Orthonormed basis in Hilbert spaces
2.Periodical function
3.Trigonometric series
4.Associated Fourier series
5.Féjer theorem (Dirichlet kernel, Féjer kernel)
6.Development in Fourier series and examples
7.Orthonormed basis in L2([a, b])
8.Other conditions of punctual and uniform convergence
9.Differentiation and integration of Fourier series
10.Theoretical aspects

CHAPTER IX. Complements of General Topology and Functional Analysis
§1 Complements of General Topology (comparison of topologies, initial and final topologies, Alexander theorem, product of topological spaces, product of Hausdorff spaces, the separation axioms (T1) – (T4), product of compact and locally compact spaces, compactifications, product of connected and locally connected spaces, axioms of countability, countably compact and sequentially compact spaces, metrisable spaces, separable spaces, Brower degree)
§2 Hahn - Banach theorem (real form, complex form, geometrical form, separation of convex sets, theorems of Dieudonné, Dunford-Schwartz, the Krein- Milman theorem, extension of linear operators)
§3 Weak topology and weak * topology (Mazur theorem, Alaoglu-Bourbaki-Kakutani theorem)
§4 Reflexive Spaces (theorems of Kakutani-Šmulian, Milman-Pettis- Kakutani, Eberlein - Šmulian , James, Krein- Šmulian, the best approximation, Day’s theorem)
§5 Topological vector spaces
1.Topological vector space
2.Locally convex space
3.Linear mappings
4.Duality
§6 Distributions
1.Test function spaces
2.Calculus with distributions
3.Localization
4.Supports of distributions
5.Convolutions
§7 Fourier transforms
1.Direct Fourier transform in L1(R)
2.Inverse Fourier transform in L1(R)
3.Direct Fourier transform in L2(R)
4.Tempered distributions
5.Paley-Wiener theorem
6.Sobolev lemma
7.Tauberian theory
§8. Laplace Transform
1.Laplace integral
2.Laplace transform
3.Laplace integral inversion
4.Representation by Laplace integral
5.Connection with Bessel functions
§9. Completions to Hilbert spaces (Elementary spectral theory)
1.Spectre of continuous operator
2.Compact operator
3.Riesz-Fredholm theory
4.Spectre of compact operator
5.Compact operators in Hilbert spaces
6.Spectral decomposition of compact self-adjoint operators