MATHEMATICAL ANALYSIS III
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- Versione italiana
- Academic year
- 2017/2018
- Teacher
- DAMIANO FOSCHI
- Credits
- 6
- Didactic period
- Primo Semestre
- SSD
- MAT/05
Training objectives
- This is the third and last fundamental course in Mathematical Analysis for students of our three-years degree. In this course we shall present the main theorems and the principal techniques of real and complex analysis, together with some applications.
The main goal of the course is to let the student know L^p spaces, Hilbert spaces, distributions, Fourier transform, letting him/her become able to work autonomously in such spaces and with such instruments.
The second goal is to let the student understand, via examples and exercises, that these spaces and instruments are widely used not only in mathematics, but also in phisics, engineering, and other sciences.
The main knowledges acquired will be:
- knowledge of L^p spaces and of the main theorems connected
- knowledge of the concept of convolution between two measurable functions
- knowledge of the concept of linear bounded operator between Banach spaces and norm of the operator
- knowledge of the concept of Hilbert space, of he main exaples of Hilbert spaces, of their properties, of the main theorems, in particular the theorem of orthogonal projections
- knowledge of the concept of distribution and of the main computations with distributions
- knowledge of the Fourier transform in L^1 and L^2, S and S' of its properties, of the inversion theorems.
The main skills acquired will be:
- ability to identify L^p functions and to operate with it, autonomously making use of instruments like Hoelder's and Minkowski's inequalities
- ability to identify a convolution and to operate with it, recognizing the support and the regularity properties of the output of a convolution
- ability to study linearity and boundedness of an operator acting between Banach spaces
- ability to recognize Hilbert spaces, to work autonomously in such spaces, in particular with orthogonal projections
- ability to recognize distributions, to perform calculations with distributions
- ability to compute the Fourier transform of a function/distribution and to use the transform's properties. Ability to solve autonomously via Fourier transform partial differential equations. Prerequisites
- Analysis I and II
Course programme
- The program of the course is the following:
- L^p spaces (14 hours).
Definition of L^p(A), A measurable subset of R^n. Conjugate exponents. Young's, Hoelder's, Minkowski's inequalities. Norm in L^p(A). Essential sup and its properties. The space L^\infty(A). Holder's inequality for {p,p'}={1,+\infty}. Inclusion theorem between L^p(A) spaces, with A of finite measure. Completeness of L^p, p[1;+\infty]. Comparison between different concepts of convergence for measurable functions: pointwise, almost everywhere, uniform, in L^p (strong). Support of a function, properties of the support, the space of continuous functions with compect support. Density of C_c in L^p and completion of C_c in L^p with respect to ||-||_p. The space C_0, completion of C_c in C_0 with respect to ||-||_\infty. Linear maps between normed spaces: continuity, boundedness, equivalence between boundedness and continuity, norm. Convolution of measurable functions, support of the convolution. L^p_{loc} spaces, their properties. Minkowski's integral inequality. Reflection, translation, Lebesgue's lemma. Convolution theorems for L^p functions.
- Hilbert spaces (12 hours).
Scalar product. Schwartz's inequality, triangle's inequality. Definition of Hilbert space. The space l^2. Parallelogram's identity. Prehilbertian spaces. Minimum norm theorem. Orthogonality. Projection theorem. Corollary. Riesz-Frechèt' s theorem. An approximation problem in Hilbert's spaces. Orthonormal sets. Bessel's inequality. Riesz-Fischer's theorem. Maximal sets. Parseval's theorem, Bessel's equality. Every Hilbert space containing an orthonormal set is isomorphic to l^2. Application to Fourier series in L^2(T).
- Fourier transform (12 hours).
Fourier transform of L^1 functions. If f is in L^1, then its transform F(f) is in C_0. Properties of the Fourier transform in L^1: behavior with respect to translation, compression/dilatation, derivation, convolution, multiplication by a power. Integration of F(f)g. Resolution of partial differential equations via Fourier transform and the problem of inversion. Inversion theorem in L^1. Injectivity of the transform. Heat equation in L^1. Wave equation in L^1. Fourier transform in L^1\cap L^2, properties. Fourier transform in L^2. Properties of the Fourier transform in L^2: norm of F(f), integration di F(f)g. Surjectivity of the transform in L^2. Inversion in L^2.
- Distributions (10 hours).
Historical introduction to the concept of distribution. The spaces D and D'. First examples of distributions: distribution functions, Dirac's delta, principal value of 1/x. Characterization theorem for distributions. Convergence for distributions, derivative of a distribution, multiplication of a distribution for a C^\infty function. The spaces S and S', Fourier transform of a distribution. Didactic methods
- The course consists in both frontal lectures and exercises. Usually, after some hours of frontal lesson introducing new concepts or important theorems, a couple of hours of exercises abouth the same argument follow. For the exercises, at the beginning of every chapter, a sheet of exercises (containing the texts of many exercises) is given to the student. He has the time to give a look autonomously to the sheet, and to try to solve some of them. In classroom the most important exercises will be solved, together with the ones explicitly asked by the students.
Learning assessment procedures
- The exam consists of two parts:
- a written examination of 3 hours, where the student is asked to solve some exercises (usually 4)
- an oral examination, consisting in a discussion about the theoretical arguments of the course.
The student is asked to discuss some of the theorems studied and of their applications.
A student is admitted to the oral examination if his mark at the written examination is greater or equal to 16. The oral examination has to be taken in the sames session as the written exam, and before the beginning of the next term's lessons.
The final mark depends on both the examinations. It isn't given by the arithmetical mean between the writtend and the oral mark, but it comes from an overall evalutation of the student's skill.
In the winter session there are at least 2 possibilities to take both the written and the oral exam, the dates are arranged with the students of the class. In the other sessions there are one or more opportunities to take the exam, depending on the studen's demand. Reference texts
- The contents of lessons and exercices mainly come from the followig books:
- W. Rudin, Analisi reale e complessa, Bollati Boringhieri
- G.Gilardi, Analisi 3, Mc Graw Hill
- F.Treves, Topological vector spaces, Accademic Press
Lessons mostly follow these lecture notes:
- L.Zanghirati, Appunti di Analisi V, in rete
- R.Agliardi, M.Cicognani, A.Corli: Esercizi di Istituzioni di Analisi Superiore, in our library.
