- Bachelor and master degrees
- TECHNOLOGY AND SCIENCE
- Degree in MATHEMATICS

TAF* | Credits number | Duration (in hours) | Period | Professors | Teaching material | |
---|---|---|---|---|---|---|

ADVANCED ANALYSIS - mod. A (529SM-1) | Programme specific subjects | 6 | 48 | Full year | Zagatti Sandro | |

ADVANCED ANALYSIS - mod. B (529SM-2) | Programme specific subjects | 6 | 48 | Full year | Del Santo Daniele | https://moodle2.units.it//course/view.php?id=6151 |

Teaching language

English

Learning objectives

By the end of the course the student will know the fundamental results of functional analysis, of metric and normed spaces theory, of Hilbert spaces theory, of the theory of AC and BV functions, of distribution theory and of the theory of Sobolev Spaces. He/she will be able to solve easy exercises and to prove autonomously easy theorems. The student will be able to recognise the classical techniques in functional analysis (duality, weak form of the

differential problems, approximation) and will be able to discuss with a certain competence on such topics. The student will be able to read the

scientific books on the discipline.

Prerequisites

Differential and integral calculus. Linear algebra.

Basic notions in measure theory. Basic notions in general topology.

Contents

Mod. A

0. Metric and normed spaces. Converging and Cauchy sequences; general properties of metric spacece. Banach spaces and space of continuous linear operators on Banach spaces.

1. Principles of Functional Analysis. Hahn-Banach theorem. Baire theorem. Banach-Steinhaus theorem. Open mapping theorem. Inverse mapping theorem. Closed graph theorem. Convexsets. Krein-Millman theorem.

2. Hilbert spaces. Inner product spaces. Bessel inequality, Schwarz inequality, polarization identity. Hilbert spaces, completion of a prehibert space, orthogonal complement, projection oeprators, direct sum. Riesz theorem. Hilbert bases, countablity of the basis and separability, Parseval identity. Lax-Milgram theorem. L^2 space, Hilbert bases in L^2. Continuous linear operators on Banach and Hilbert spaces, adjoint operator, selfadjoint operators.

3. Space of continuous functions on a compact metric space. Completeness and separability. Partition of unity. Stone-Weiertrass theorem. Equicontinuity and Ascoli-Arzelà theorem.

4. Topology generated by a family of functions. Weak topology in a Banach space. Mazur lemma. Bidual space and reflexive spaces. Strong and weak continuity of linear operators. Weak* topology. Banach-Alaoglu theorem. Helly lemma. Kakutani theorem. Weierstrass theorem on the minimum of a

sequential weakly lower semicontinuous functional. Uniform convexity and Millman theorem, weak-strong convergence in a uniform convex

space.

5. L^p spaces. Holder inequality, Minkowsky inequality, separability, interpolation, Clarkson inequality and uniform convexity, reflexivity. Duality and Riesz theorem. Convolution, Young inequality, function with compact support, mollifiers, strong compactness criterion. Weak convergence.

6. Weak differentiation. Definition of Sobolev space and their main properties: completeness, separability, uniform convexity and reflexivity.

Mod. B

1. AC and BV functions. Nowhere differentiable functions. Lebesgue theorem on differentiability a.e. of monotone functions. Functions with bounded variation (BV). Absolutely continuous functions (AC), fundamental theorem of calculus in Lebesgue framework.

2. Differentiation of measures. Hanh decomposition theorem, Radon-Nikodim Theorem. Hardy-Littlewood maximal function. Symmetric derivative of a measure, theorem on Lebesgue points for L^1 functions.

3. Distributions. Test functions. Distributions of finer and infinite order. Derivatives in the sense of distributions. The "Théoreme de structure". Distributon with compact support. Convolution of distributions. Fourier transform of L^1 functions. Schwartz space, Temperate distributions. Fourier transform of temperate distributions. Plancherel theorem. Fourier-Laplace transform of a distribution with compact support. Paley-Wiener theorem.

4. Sobolev spaces in 1 D. Definition and characterization of Sobolev Spaces in 1 D. Results of extension and density. Sobolev embeddings, Rellich theorem. Poincaré inequality. Boundary value problems in 1D. Maximum principle in 1D.

5. Sobolev spaces in N D. Friedrichs lemma. BV functions in N D. Extension of Sobolev functions defined in open sets. Sobolev-Gagliardo-Nirenberg theorem, Morrey theorem. Rellich theorem. Poincaré inequality in N D. Boundary value problems in N D.

Teaching format

Lectures and exercises in class. A part of the teaching material will be put on Moodle. Some home-works will be proposed and discussed in a second time in class.

Extended Programme

Mod. A

0. Metric and normed spaces. Converging and Cauchy sequences; general properties of metric spacece. Banach spaces and space of continuous linear operators on Banach spaces.

1. Principles of Functional Analysis. Hahn-Banach theorem. Baire theorem. Banach-Steinhaus theorem. Open mapping theorem. Inverse mapping theorem. Closed graph theorem. Convexsets. Krein-Millman theorem.

2. Hilbert spaces. Inner product spaces. Bessel inequality, Schwarz inequality, polarization identity. Hilbert spaces, completion of a prehibert space, orthogonal complement, projection oeprators, direct sum. Riesz theorem. Hilbert bases, countablity of the basis and separability, Parseval identity. Lax-Milgram theorem. L^2 space, Hilbert bases in L^2. Continuous linear operators on Banach and Hilbert spaces, adjoint operator, selfadjoint operators.

3. Space of continuous functions on a compact metric space. Completeness and separability. Partition of unity. Stone-Weiertrass theorem. Equicontinuity and Ascoli-Arzelà theorem.

4. Topology generated by a family of functions. Weak topology in a Banach space. Mazur lemma. Bidual space and reflexive spaces. Strong and weak continuity of linear operators. Weak* topology. Banach-Alaoglu theorem. Helly lemma. Kakutani theorem. Weierstrass theorem on the minimum of a

sequential weakly lower semicontinuous functional. Uniform convexity and Millman theorem, weak-strong convergence in a uniform convex

space.

5. L^p spaces. Holder inequality, Minkowsky inequality, separability, interpolation, Clarkson inequality and uniform convexity, reflexivity. Duality and Riesz theorem. Convolution, Young inequality, function with compact support, mollifiers, strong compactness criterion. Weak convergence.

6. Weak differentiation. Definition of Sobolev space and their main properties: completeness, separability, uniform convexity and reflexivity.

Mod. B

1. AC and BV functions. Nowhere differentiable functions. Lebesgue theorem on differentiability a.e. of monotone functions. Functions with bounded variation (BV). Absolutely continuous functions (AC), fundamental theorem of calculus in Lebesgue framework.

2. Differentiation of measures. Hanh decomposition theorem, Radon-Nikodim Theorem. Hardy-Littlewood maximal function. Symmetric derivative of a measure, theorem on Lebesgue points for L^1 functions.

3. Distributions. Test functions. Distributions of finer and infinite order. Derivatives in the sense of distributions. The "Théoreme de structure". Distributon with compact support. Convolution of distributions. Fourier transform of L^1 functions. Schwartz space, Temperate distributions. Fourier transform of temperate distributions. Plancherel theorem. Fourier-Laplace transform of a distribution with compact support. Paley-Wiener theorem.

4. Sobolev spaces in 1 D. Definition and characterization of Sobolev Spaces in 1 D. Results of extension and density. Sobolev embeddings, Rellich theorem. Poincaré inequality. Boundary value problems in 1D. Maximum principle in 1D.

5. Sobolev spaces in N D. Friedrichs lemma. BV functions in N D. Extension of Sobolev functions defined in open sets. Sobolev-Gagliardo-Nirenberg theorem, Morrey theorem. Rellich theorem. Poincaré inequality in N D. Boundary value problems in N D.

End-of-course test

Mod. A

The final exam is in two parts. The written part (3-4 hours) consists in the solution of some problems concerning the contents of the course. The

oral part is devoted to ascertain the comprehension and the managing of the topics reached by the candidate.

Mod. B

Final oral examination on the whole content of the course. Also some easy exercises (solved during the oral discussion) will be possibly part of the examination.

Other information

Teachers receive students by email appointment (details are avaible in the personal web-pages of teachers).

The course will possibly host lessons of foreign professors in the framework of an Erasmus exchange.

Any possible change to what described here, which will become necessary to ensure the application of the safety protocols related to the COVID19 emergency, will be communicated on the Department, Study Program and teaching website.

Texts/Books

- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer.

Units Libray position

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0345510

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0053295

- E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer.

Units Libray position

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0608477

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0271866

- F. Hirsch, G. Lacombe, Elements of functional analysis, Springer.

Units Libray position

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA2349555

- L. Hörmander, Linear Partial Differential Operators, Spinger.

Units Libray position

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0714367

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0262727

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0208945

- A. N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover.

Units Libray position

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0549953

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA2340678

- M. Reed, B. Simon, Functional Analysis, Academic Press.

Units Libray position

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0220742

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0235874

- W. Rudin, Real and Complex Analysis, McGraw-Hill.

Units Libray position

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0238135

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0338820

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0053401

- K. Yosida, Functional analysis, Springer.

Units Libray position

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0090679

https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0248368