Course Details
Name:
Istituzioni Di Analisi Superiore / Istituzioni Di Analisi Superiore
Basic information
Credits:
Bachelor Degree in Mathematics: 12 Ects (b)
Term:
Module Complex analysis (Istituzioni di Analisi Superiore mod. 2): 2° semester
Module Functional analysis: 1° semester
Degree(s):
Compulsory 3^{rd} year Bachelor Degree in Mathematics curriculum Generale
Language:
Italian
Course Objectives
Module Complex analysis (Istituzioni di Analisi Superiore mod. 2): Knowledge of basic topics of complex analysis: elementary functions of complex variable, differentiation, integration and main theorems on analytic functions . Ability to use such knowledge in solving problems and exercises
Module Functional analysis: Learn the fundamental structures of Functional Analysis. Get familiar with the main examples of functional spaces, in particular with the theory of Hilbert spaces and Lebesgue spaces. Get familiar with the basic notions of operator theory. Be able to frame a functional equation in an abstract functional setting.
Course Content
Module Complex analysis (Istituzioni di Analisi Superiore mod. 2)
- Complex numbers. Sequences. Elementary functions of complex numbers. Limits, continuity. Differentiation. Analytic functions. Armonic functions
- Contour integrals. Cauchy's Theorem. Cauchy's integral formula. Maximum modulus theorem. Liouville's theorem. Morera theorem.
- Series representation of analytic functions. Taylor's theorem. Laurent's series and classification of singularities
- Calculus of residues. The residue theorem. Application in evaluation of integrals on the real line and Principal Value. The logarithmic residue, Rouche's theorem.
- Fourier transform for L^1 functions. Applications. Fourier transform for L^2 functions. Plancherel theorem.
- Laplace transform and applications.
Module Functional analysis
- Lebesgue Measure and Integration
- L^p Spaces
- Basic of Topological Vector Spaces, Normed and Banach Spaces, Linear Operators and linear functionals.
- Hilbert Spaces
- Weak topology, Weak * topology, weak compactness
- Applications of Baire Category in Functional Analysis: Uniform Boundedness, Open Mapping,
Closed Graph, Inverse Mapping.
- Banach and Hilbert adjointness, self-adjointness
- Compact Operators
- Riesz Fredholm spectral theory
Learning Outcomes (Dublin Descriptors)
On successful completion of this course, the student should
Module Functional analysis
- Understand the theory
- Be able to solve problems
- Help to choose appropriate graduate studies
- Practice mathematical reasoning, organize topics in logical order, connect theory to applications,
elaborate independent proofs. Improve unconventional thinking.
- Get the math language to study more advanced textbooks and attend research oriented courses
Prerequisites and Learning Activities
Module Complex analysis (Istituzioni di Analisi Superiore mod. 2): Knowledge of all topics treated the Mathematical Analysis courses in the first and second year : real function of real variables, limits, differentiation, integration; sequences and series of funcions; ordinary differential equations
Module Functional analysis: Mathematical Analysis (not only Calculus) in one and several space variable, Linear Algebra
(including abstract Vector Speces), Set Topology (including Compact Spaces) and
Metric Spaces, Ordinary differential equations.
Teaching Methods
Language: Italian
Module Complex analysis (Istituzioni di Analisi Superiore mod. 2): theoretical lectures and exercises
Module Functional analysis: Classical traditional XX century blackboard teaching (no fancy modern technology)
Assessment Methods and Criteria
Module Complex analysis (Istituzioni di Analisi Superiore mod. 2): Written exam and oral exam
Module Functional analysis: Written test.
Textbooks
Module Complex analysis (Istituzioni di Analisi Superiore mod. 2)
- J.E. Marsden, M.J. Hoffman, Basic complex analysis. Freeman New York.
- W. Rudin, Real and complex analysis. Mc Graw Hill.
Module Functional analysis
- Terence Tao, An introduction to measure theory.. American Mathematical Society, Providence, RI, ISBN: 978-0-8218-6919-2 . 2011.
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations.. Universitext. Springer, New York,. 2011. xiv+599 pp. ISBN: 978-0-387-70913-0
- Alberto Bressan, Lecture notes on functional analysis. With applications to linear partial different. Graduate Studies in Mathematics, 143. American Mathematical Society, Providence, RI,. 2013. xii+250 pp. ISBN: 978-0-8218-8771-4
- Lecture notes provided by the teacher.
- Michael Reed, Barry Simon, Methods of modern mathematical physics. I. Functional analysis. Second edition. . Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York,. 1980. xv+400 pp. ISBN: 0-12-585050-6
- Stein, Elias M.; Shakarchi, Rami , Real analysis. Measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, III.. Princeton University Press, Princeton, NJ,. 2005. xx+402 pp. ISBN: 0-691-11386-6
Online Teaching Resources
Course page updates
This course page is available (with possible updates) also for the following academic years:
Course information last updated on: 08 ottobre 2018, 14:39