Lieb-Loss, Analysis, Graduate Studies in Mathematics, AMS, Providence, RI, USA.
Rudin, Analisi Reale e Complessa, Boringhieri, Torino.
Brezis, Analisi Funzionale, Liguori Editore, Napoli.
Evans, Partial Differential Equations, Graduate Studies in Mathematics, AMS, Providence, RI, USA.
Garabedian, Partial Differential Equations, AMS Chelsea Publishing, Providence, RI, USA.
Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, AMS, Providence, RI, USA.
Rudin, Functional Analysis, McGraw Hill.
T. Tao, An epsilon of room, AMS.
R. Magnanini, Lecture notes (in Italian), unloadable from http://web.math.unifi.it/users/magnanin/Istit/2010.gsm.disp.pdf
W. Rudin, Functional analysis, McGraw Hill.
Learning Objectives
Knowledge acquired:
The fundamental ideas of functional analysis in Hilbert spaces, of the theory of convex sets and functions, of harmonic analysis, the corresponding techniques of proof and some of their relevant applications.
Competence acquired:
The basic methods of proof in functional and harmonic analysis and the connections of this topics to other disciplines of mathematics.
Skills acquired (at the end of the course):
The student is expected to correctly state, prove and apply the main results of functional and harmonic analysis and the theory of convex functions, and to solve exercises and problems related to those disciplines.
Prerequisites
Courses to be used as requirements (required and/or recommended) Courses required: Mathematical Analysis 1, 2 and 3; Geometry 1 and 2. Courses recommended:
Teaching Methods
CFU: 9 Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 225 Hours reserved to private study and other indivual formative activities: 153 Hours for lectures: 72 Hours for laboratory: 0 Hours for laboratory-field/practice: 0 Seminars (hours): 0 Stages (hours): 0 Intermediate examinations (hours): 0
Further information
Attendance of lectures, practice and lab: Not mandatory
Type of Assessment
Discussion of main topics of the course and final oral examination in which the student must be able to correctly state, prove and apply the main results of functional and harmonic analysis, basic theory of convex functions and sets, and to solve exercises and problems related to those topics
Course program
Supplements of Real Analysis.
Vitali's covering Lemma
Functions of bounded variations
Absolutely continuous functions
Fundamental theorem of calculus for absolutely continuous functions
Derivatives in the sense of distribution and Sobolev spaces.
FUNCTIONAL ANALYSIS IN HILBERT SPACES
• Definition of dot product, Cauchy-Schwarz inequality, the parallelogram identity.
• Definition of Hilbert space and examples.
• Projection on a convex set.
• Orthonormal systems. Series and Fourier coefficients. Bessel's inequality.
• Completeness of orthonormal systems. Parseval identity.
• Linear and bounded functionals. Dual space. Riesz representation theorem. Weak convergence.
Banach-Alaoglu's theorem.
• The theorems of Stampacchia and Lax-Milgram.
• Open mapping theorem and its consequences.
• Banach-Steinhaus's theorem.
• Linear and continuous operators.
• Compact operators: examples and first properties.
• Fredholm alternative's theorem.
• Spectrum of a compact operator.
• Spectral decomposition of a symmetric and compact operator.
Fourier series
• Trigonometric polynomials.
• Series and Fourier coefficients.
• Bessel's inequality.
• Riemann-Lebesgue's lemma.
• Dirichlet’s kernel.
• Dini's and Jordan's criteria.
• Uniform convergence of Fourier series.
• Completeness of trigonometric system in the space of square integrable functions.
• Parseval identity.
Fourier Transform
• Fourier transform of a summable function.
• Behavior of Fourier transform with respect to dilation, translation, rotation and convolutions.
• The space of Schwarz. Derivation and transform.
• Transform of a Gaussian.
• Plancherel theorem and Parseval identity. Definition of Fourier transform in the space of square integrable functions.
• Inversion formula.
Basic theory of convex sets and functions
• Convex functions of one variable, basic properties, monotonicity of the incremental ratio, existence of first derivative
• Non existence of local maxima, second order incremental ratio and existence of second derivative
• Convex sets in R^n. Convex combinations and characterization of convexity in their terms. Convex envelope.
• Caratheodory Theorem. Extremal points, Minkowski Theorem. Separation theorems and supporting hyperplanes.
• Convex functions of several variables. Jensen inequality. Maximun attained in extreme points. A convex function on a polytope is bounded. Being Lipschitz in the interior of the domain
• Existence of support functions, subgradient. Sublinear functions and directional derivative. Existence of all directional derivatives and differentiability. Differentiability almost everywhere.
HARMONIC FUNCTIONS
• Mean value property.
• Harmonic functions are infinitely differentiable.
• Maximum principle.
• Hopf principle.
• Harnack's inequality.
• Liouville's theorem.
• Harnack's convergence theorems.
BOUNDARY Value PROBLEMS FOR LAPLACE AND POISSON's EQUATIONS
• Poisson’s equation in Euclidean space and its resolution.
• Fundamental solution for Laplace's equation.
• Uniqueness theorems for Dirichlet, Neumann and Robin problems.
• Boundary value problems in unbounded domains. Kelvin transformation.
• Stokes identity and definition Green’s function.
• Representation formula for the solution of Dirichlet's problem.
• Symmetry of Green’s function.
• Construction of Green's function and Poisson's kernel for half space and sphere.
• Analyticity of harmonic functions.
• Subharmonic and superarmonic functions.
• Perron's method.
• Barrier functions. Regular and exceptional points.
• Existence of Green’s function.
• Dirichlet's principle and Lax-Milgram's theorem.
• Conversion Dirichlet or Neumann problem into an integral equation and application of Fredholm's alternative theorem.
• Eigenvalues and eigenfunctions of Laplace's operator.