Constructive Aspects of Functional Analysis: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Erice ... 27-July 7, 1971 (C.I.M.E. Summer Schools, 57)

Cover
Title
Contents
A Constructive Approach to Optimal Control
	0. Introduction
	1. Relaxed Controls
	2. The basic Technique
	3. Fixed End-Point Problems
	References
	Lecture Notes II
		Stochastic Systems
			1. Inducing Measures on C; the Wiener measure
				References
			Example: Linear Stochastic Control
				References
		Identification and Adaptive Control: An Example
			1. Basic Dynamics and Measurements
			2. System Identification
			3. Control Theory
		Appendix I Identification Theory
		Appendix II
			Nomenclature
		Appendix III
			References
Methodes Iteratives Duales Pour La Minimisation De Fonctionnelles Convexes
	1. Introduction
	2. Cadre fonctionnel
	3. Exemples
	4. Un algorithme de recherche de point-selle
	5. Un deuxième algorithme de recherche de point-selle
	6. Application du premier algorithme à l\'exemple 3.1
	7. Considerations sur la minimisation par dualite de fonctionelles convexes non differentiables
		7.1 : Generalites
		7.2 : Position de problème
		7.3 : Determination d\'un problème dual
	8. Un problème de contrôle optimal avec fonction cout non differentiable
	9. Une remarque sur le comportement des multiplicateurs de Lagrange
	Bibliographie
Approximation Numerique Des Inequations D\'evolution
	Introduction
	Chapitre I. Inequations d\'evolutions paraboliques Type I
		1. Exemples
		2. Formulation generale
		3. Solutions fortes et faibles
		4. Generalites sur les methodes constructives d\'approximation
			4.1. Reduction à une equation parabolique. Penalisation
			4.2. Reduction à un equation parabolique. Regularisation
			4.3 Reduction à un inequation elliptique. Regularisation elliptique
			4.4 Reduction à une inequation elliptique. Discretisation
			4.5 Inequations d\'evolution et points selles(cf. Tremolièreees [1])
	Chapitre 2. Approximation par discretisation des inequations paraboliques des inequations paraboliques de type I
		1. Approximation d\'un couple d\'espaces Constante de stabilite
		2. Schemas d\'approximation des inequations paraboliques de type I
		3. Analys de la stabilite
		4. Etude de la convergence
	Chapitre 3. Inequations d\'evolution paraboliques de type II
		1. Exemples
		2. Formulation generales
		3. Schemas d\'approximation
		4. Stabilite et convergence (cf. D Viaud |1|)
	Chapitre 4. Inequation d\'evolution du 2eme ordre en t
		1. Exemples
		2. Formulation generale
		3. Schemas d\'approximation
		4. Stabilite et convergence
	Chapire 5. Complements et problèmes
		1. Ecoulement de fluides de Bingham bidimensionnels
		2. Problèmes ouverts
	Bibliographie
Introduction into the Methods of Numerical Analysis
	Introduction
	Chapter 1 General Information from the Theory of Difference Schemes
		1.1 Basic and Adjoint Equations
		1.2 Approximation
		1.3 Numerical Stability
		1.4 The Convergence Theorem
	Chapter 2 Methodes of Solution of Nonstationary Problems
		2.1 Approximation - Stability Relation
		2.2 Difference schemes of second-order accuracy with time-de-pendent operators
		2.3 Inhomogeneous evolution equations
		2.4 Splitting-up methods for nonstationary problems
		2.5 Multi - component splitting of problems
		2.6 A General Approach to Component-wise Splitting
		2.7 Hyperbolic Equations
	References
		1. Monographs and text books
		2. Additional Literature
An Introduction to the Approximate Solution of Variational Inequalities
	Chapter I Minimum problems and variational inequalities: Convexity, monotonicity and fixed-points
		1. Direct formulation
		2. Weak formulation
		3. The linearized formulation
		4. The fixed-point formulation
		5. The epigraph formulation
		6. Minimum problems in normed spaces
		7. Monotone operators and variational inequalities: the linearization lemma
		8. Variational inequalities and fixed-points
		9. Minimization of non-differentiable functionals and mixed variational inequalities
	Chapter 2
		1. Some variational inequalities
		2. Finite-dimensional and iterative existence theorems
		3. Variational inequallities for bilinear forms in Hilbert spaces
		4. Direct existence theorem
	Chapter 3 Convergence of convex sets and of solutions of variational inequallities
		1. The Ritz-Galerkin approximation
		2. A convergence for convex sets and convex functions
		3. The \"stability\" theorem
		4. Further exitence theorems
		5. Finite-dimensional approximation I: the discrete problem
		6. Finite-dimensional approximation, II: convergence of the approximate solutions
		7. Dual variational inequalities and complementarity systems
		8. An example
	References
Best approximation in normed linear spaces
	Introduction
	1. Characterizations of elements of best approximation
		1.1 The first main theorem of characterizatiion
		1.2 The second main theorem of characterization
		1.3 Differential characterizations
	2. Existence of elements of best approximation
		2.1 Characterizations of proximinal linear subspaces
		2.2 Some classes of proximinal linear subspaces
		2.3 Normed linear spaces in which all closed linear subspaces are proximinal
		2.4 Normed linear spaces which are proximinal in every super-spaces
		2.5 Transitivity of proximinality
		2.6 Proximinality and quotient spaces
	3. Uniqueness of elements of best approximation
		3.1 Characterizations of semi-Cebysev and Cebysev subspaces
		3.2 Existence of semi-Cebysev and Cebysev subspaces
		3.3 Normed linear spaces in which all linear subspaces are semi-Cebysev
		3.4 Semi-Cebysev and Cebysev subspaces and quotient spaces
		3.5 Strongly unique elements of best approximation
		3.6. Almost Cebysev subspaces
	4. Properties of metric projections
		4.1 Definition and some properties of metric projections
		4.2 Continuity of metric projections
		4.3 Weak continuity of metric projections
		4.4 Lipschitzian metric projections
		4.5 Differentiability of metric projections
		4.6 Linearity of metric projections
		4.7 Semi-continuity and continuity of set-valued metric projections
		4.8 Continuous selections and linear selections for set-valued metric projections
	5. Best approximation by elements of non-linear sets
		5.1 Best approximation by elements of convex sets
		5.2 Best approximation by elements of N-parameter sets
		5.3 Generalizations
		5.4 Best approximation by elements of arbitrary sets
	References
A Fourier Analysis of the Finite Element Variational Method
	0. Apologia
	1. Introduction
	2. Approximation
	References
Caracteristiques D\'approximation Des Compacts Dans Les Espaces Fonctionnels Et Problemes Aux Limites Elliptiques
	Bibliographie