Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
Academic year in which the course will be held: 
Course type: 
Compulsory subjects, characteristic of the class
Second semester
Standard lectures hours: 
Detail of lecture’s hours: 
Lesson (64 hours)

The content of Mathematical Analysis 1-3, Linear Algebra and Geometry, Geometry 1

Final Examination: 

- Homework exercises which will verify the acquisition of an operational knowledge of the subject and the ability to apply the techniques illustrated in class to produce indpendently proofs of statements similar to those seen in in the lectures and to express themselves in rigorous mathematical langauge.

- Final oral exam devoted to the discussion of the homework exercises, and to the proof of one or two theorems seen in class. This part will assess the acquisition of an in-depth knowledge of the topics presented in class and the students' ability to express themselves in a rigorous mathematical language and to recognize the validity of, even subtle, mathematical reasonings.

Voto Finale

The course is the natural continuation of the sequence Mathematical Analysis 1-3, and it aims to deepen the study of classical and modern analysis begun in the previous courses.
Students will acquire a working knowledge of the methods of advanced analysis. They will know statements and proofs of the main theorems, and will be able to solve exercises, even of theoretical nature, about the topics dealt with in the course. They will have learned a number of techniques of proof which they will be able to use to recognize the validity of mathematical reasoning, even rather sophisticated, and to prove results related to those described in class. Finally students will be able to express themselves in a rigorous mathematical language.

Hilbert spaces. Orthogonality and orthonormal bases. Riesz representation theorem and Hilbert space duals. Orthonormal bases in L2(-\pi,\pi). Trigonometric polynomials and Fourier series on the torus. L2
theory: Bessel inequality and Parseval and Plancherel identities Polinomi trigonometrici e serie di Fourier sul toro. Teoria L2: Bessel inequality and Parseval and Plancherel identities. Pointwise convergenge. Isoperimetric inequality in R2.
Lebesque differentiation: differentiation of monotone functions, functions of bounded variation, differentiation of an integral and absolutely continuous functions, characterization of absolutely continuous functions.
Topics in measure theory: Borel measure and regularity property. The Riesz representation theorem. Luzin theorem. Signed and complex measures: total variation and the Hahn and Lebesque decomposition theorems. The Radon-Nykodym theorem. Duals of the Lp spaces.
Convolution in Rn, Minkoswki integral inequality and Young’s Theorem. Regularization kernels. Introduction to Hausdorff measure.

- E. Giusti, Analisi Matematica 2, Boringhieri
- G. Folland, Real analysis: modern techniques and their applications, Wiley
- H. Royden, Real Analysis, Mc Millan
- W. Rudin, Real and Complex Analysis, Mc Graw Hill.
- E. Stein and R. Shakarchi, Real Analysis, PUP
- R. Wheeden and A. Zygmund, Measure and integral: an introduction to real analysis, CRC

Frontal lectures: 64 hours

Office hours: by appointment (email the instructor)

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