Unit MATHEMATICAL ANALYSIS IV
- Course
- Mathematics
- Study-unit Code
- 55078209
- Curriculum
- In all curricula
- Teacher
- Anna Rita Sambucini
- Teachers
-
- Anna Rita Sambucini
- Hours
- 63 ore - Anna Rita Sambucini
- CFU
- 9
- Course Regulation
- Coorte 2020
- Offered
- 2022/23
- Learning activities
- Caratterizzante
- Area
- Formazione teorica
- Academic discipline
- MAT/05
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- The aim of the course is to give basic properties on linear Real Analysis and to present problems and applications that come from the physical, biological, chemical and economical phenomena.
- Reference texts
- P. Cannarsa & T. D'Aprile, Introduzione alla teoria della misura e all'analisi funzionale, UNITEXT, Springer, 2008, xii+268 pp.
R.G. Bartle, The elements of integration and Lebesgue measure, Wiley Classics Library, Wiley-Interscience Publ., New York, 1995, xii+179 pp.
F.S. Botelho, Real analysis and applications, Springer, Cham, 2018. xiii+567 pp.
H.D. Junghenn, Principles of analysis. Measure, integration, functional analysis, and applications, CRC Press, Boca Raton, FL, 2018, xx+520 pp.
The slides of the lessons are published weekly on Unistudium - Educational objectives
- The student should acquire a basic knowledge in real analysis as well as in Lebesgue and Hilbert spaces theory. The course matter is part of the contents of a standard reformed second level course for Italian three-year degrees in Mathematics. Even the setting is reformed, and the textbooks used are rich of examples and counterexamples, and therefore seem to be optimal to achieve a good understanding of definitions and statements of theorems.
The course aims at analyzing the basic arguments of real analysis and of functional analysis in Hilbert spaces, at treating so widespread and comprehensive discipline as taught for years at national and international levels. In this sense, the purpose of the course is to make the students able
- To know the main topics of real analysis and integration theory and how to apply them to the natural sciences,
- To own computational skills to solve various exercises,
- To read and understand texts of Real Analysis and Functional Analysis,
- To provide themselves a mathematical proof of simple statements,
- To communicate in Italian the mathematical knowledge acquired in the course, as well as related issues,
- To work in teams, but also in autonomy.
The skills listed above are set out in the framework of the professions related to both a traditional mathematician, and a mathematician oriented to technical and/or industrial activities. - Prerequisites
- To better understand the topics covered in the course the student should have passed the exams of Mathematical Analysis I, II and III. In particular, the course aims at making the student familiar with the theories that play a central role in modern mathematics, such as integration and functional analysis in Hilbert spaces. Therefore, the prerequisites are concepts that students meet not only in basic courses of Mathematics but, increasingly, also in their pre-university education.
- Teaching methods
- The course is split into traditional lectures, in which several exercises are carried out in class. The essential arguments are summarized in handouts provided by the teacher. The course is divided into 63 hours of theory, together with different examples and counterexamples (almost 20 hours are dedicated to practical exercises). In the tutorial service the students will be followed individually by the teacher.
To better understand the topics covered in the course the student should have passed the exams of Mathematical Analysis I, II and III. In particular, the course aims at making the student familiar with the theories that play a central role in modern mathematics, such as integration and functional analysis in Hilbert spaces. Therefore, the prerequisites are concepts that students meet not only in basic courses of Mathematics but, increasingly, also in their pre-university education. - Other information
- The teacher will distribute educational materials useful for a better understanding of the course, in order to help and to let the students pass easily the exam. As an experiment, the course could be done wholly or partly in English, with the agreement of the students attending it. In any case, the oral exam may be conducted in the English language at the request of the student.
The course is divided into 63 hours of theory, together with different examples and counterexamples (almost 20 hours are dedicated to practical exercises). In the tutorial service the students will be followed individually by the teacher.
To better understand the topics covered in the course the student should know the basic topics of Mathematical Analysis I, II and III acquired in previous courses in Mathematical Analysis of the Laurea Triennale in Matematics. In particular, the course aims at making the student familiar with the theories that play a central role in modern Mathematics, such as Real Analysis, with its use in applications.
The lecture attendance, even if not mandatory and compulsory, is strongly recommended for a better understanding of the topic. - Learning verification modality
- The exam includes one test with the performance of some exercises. The oral exam consists of a discussion on three topics one of which divided into several questions and takes about 30 minutes. The oral test is designed to assess the level of knowledge attained by the student on the theoretical contents and on the methodologies of the course (fundamental theorems, definitions, examples and counterexamples). Finally, the oral examination allows the teacher to verify the performance of the student and his/her ability to organize the presentation in autonomy.
The examination timetable is divided into 8 exam sessions and the examination dates are available at the web page http://www.dmi.unipg.it/didattica/corsi-di-studio-in-matematica/matematica-triennale/calendario-esami
Students with disabilities must inform the teacher of their status with a note when booking the exam (at least one week before) in order to allow for an appropriate organization of the written test - Extended program
- Lebesgue spaces: definition, completeness, separability, duality. Theorems of limits under the sign of integrals. Convergences: in measure, quasi-uniform. The theorem of Vitali and comparison of variuos notions of convergence. Functions of bounded variation and absolutely continuous functions: differentiability and integrability properties. Hilbert spaces: Euclidean spaces, parallelogram identity, projection theorem, duality, orthonormal systems, trigonometric series. Strong convergence theorms in Lp(X). Dense subsets of Lp(X).