B018815 - MATHEMATICAL ANALYSIS III

Italian

Lebesgue measure. Vitali and Cantor sets. Egoroff and Lusin theorems.
Lebesgue integral. Limit, derivation and integral. Fubini's theorem.
L^p spaces. Completeness and separability. Dual and weak convergence.
Convolution. Young's inequality. Mollifiers.
Hilbert spaces. Projections. Orthogonal. Bases. Linear functionals. Dual norm. Riesz's representation theorem.
Ascoli-Arzelà Theorem.
Holomorphic function and properties. Meromorphy, essential singularities. Residue Theorem.

Christopher Heil, Introduction to Real Analysis, Springer
R. Magnanini, Dispense di Analisi Matematica III, scaricabili da http://web.math.unifi.it/users/magnanin/Istit/a3gsm15.pdf

Knowledge acquired: The fundamental ideas of real analysis; in particular, the notions of measure of sets and integral based on the principles established for the theory of Lebesgue at the beginning of the XX century. Basic ideas of functional analysis such as the concept of orthogonality in Hilbert spaces and the topologic and geometric properties of Lebesgue spaces of functions with summable power. The basic properties of olomorphic functions of one complex variable.
Competence acquired: The student should be familiar with the proposed techniques, such as convergence theorems; the theorems of Fubini, Ascoli-Arzelà, Riesz, Banach-Alaoglu, Cauchy's formula, the theorem of residues, and some of their applications.
Skills acquired (at the end of the course):
The student is expected to correctly state, prove and apply the main results of real and complex analysis and the theory of Lebesge spaces, and to solve exercises and problems related to those issues.

Courses required: Mathematical Analysis I and II, Geometry I.

CFU: 9
Number of lecture hours: 70
Number of hours devoted to practicing problems and exercises in class: 14

Frequency of lectures, practice and lab: Recommended
Teaching Tools Moodle Class https://e-l.unifi.it/course/view.php?id=28421

Intermediate and final written examination: A selection of problems is proposed. The tests are designed to assess the ability of the students to apply their skills and their knowledge. In the evaluation, special attention is paid to the correctness of the solution procedure, as well as to the originality and effectiveness of the methods adopted. Obtaining a sufficient grade in the written text is necessary for participating to the oral examination.

Oral examination: A number of questions are posed. The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. In the assessment, special attention is paid to communication skills, critical thinking and appropriate use of mathematical language.

Measure theory and Lebesgue integration.
Measure of open and compact sets in euclidean space and its properties.
Lebesgue measurable sets and their measure.
Main properties of measurable set (numerable union, intersection, difference).
Countable additivity of Lebesgue's measure.
Sequences of measurable sets.
Examples: Lebesgue measurable set not Peano-Jordan measurable; Cantor’s set and function; open set of arbitrarily small measure with boundary of infinite measure; Vitali’s set (not Lebesgue measurable).
Measurable spaces, sigma-algebras, positive measures.
Examples of measures: counting measure, Dirac's delta.
Measurable functions and their properties.
Semi-continuous functions.
Simple functions.
Approximation of measurable functions by simple functions.
Integral of a non-negative measurable function and its elementary properties.
Beppo-Levi’s theorem on monotone convergence.
Fatou’s lemma.
Countable additivity of the integral of non-negative function.
Integral of summable functions and its properties.
Absolute continuity of the integral.
Lebesgue’s theorem of dominated convergence.
Comparison between Riemann’s and Lebesgue’s integral.
Fubini's and Tonelli's theorems.
Carathéodory's theorem.
Hausdorff's measure.
Definition of Hilbert space.
Orthonormal systems.
Bessel's inequality.
Hilbert's bases and Parseval's identity.
Some basics on convex functions.
Convex and concave functions of one real variable and their properties with respect to limit and order.
Monotony of the incremental ratio of a convex function.
Derivatives of convex functions.
Support line.
Jensen’s and Young's inequalities. Arithmetic and geometric means.
Lp spaces.
Holder’s inequality.
Minkowsky’s inequality.
Essential supremum and space of essentially bounded functions.
Lp is a linear normed vector space.
Lp is complete.
Definition of space of Banach and of Hilbert.
Norm differentiability.
Projection on closed and convex sets.
Linear and continuous functionals on Lp and weak convergence.
Linear functionals separate.
Lower semi-continuity of the norm.
The dual space of Lp: Riesz's representation theorem.
Convolutions. Young’s inequality with q=1 and p=r.
Approxymation by simple and C-infinite functions with compact support.
Support of a measurable function.
Separability of Lp.
The bounded sets of Lp are weakly compact.
Convolutions of functions in dual spaces are continuous.
The spaces L1 and L-infinite.
Arzelà's theorem.

Convergence comparisons:
Egorov-Severini’s theorem.
Lusin’s theorem.
Convergence in measure.
Comparisons of convergences: in measure, almost everywhere, strongly in Lp and weakly in Lp.

Definition of olomorphic function of one complex variable.
Cauchy-Riemann equations.
Elementary olomorphic functions.
Cauchy, Morera and Goursat's theorems.
Cauchy formula.
Analiticity of olomorfhic functions and unique continuatio principle.
Maximum modulus theorem.
Laurent's series.
Theorem of residues and argument's principle.
Computation of improper integrals by the theorem of residues.