English

By the end of the course the student will be able to manage the fundamental tools of Functional Analysis and to approach the further steps of this area of Mathematics, like the theory of Distributions and of Sobolev spaces, in order to face the first arguments in Partial Differential Equations and Calculus of Variations. The student will be able to read autonomously advanced monographs of Functional Analysis, to understand proofs and applications of theorems of any level, and to apply them to various fields of Mathematics. He/she will be able to solve problems of basic level.

Calculus I and II. Basic notions in measure theory. Basic notions in general topology.

a) Fundamental principles of Functional Analysis: Banach and Hilbert

spaces. Continuous linear operators.

b) Basic function spaces: continuous functions, Lebesgue spaces (L^p).

c) Weak topologies.

Lectures and solutions of problems.

0. Metric and normed spaces. Converging and Cauchy sequences;

completeness, compactness, precompactness and relative compactness;

density and separablity. Banach spaces and space of continuous linear

operators on Banach spaces, dual space and extension theorem.

1. Analytic form of Hahn-Banach theorem, gauge of a convex set, first

and second geometric form of Hahn-Banach theorem. Baire theorem,

Banach-Steinhaus theorem, open mapping theorem, inverse mapping

theorem, closed graph theorem. Diagonal procedure. Convex

sets and convex functions. Examples: space of continuos functions on a real interval and

L^p spaces.

2. Inner product spaces, orthogonality and orthonormality, orthonormal

systems. Pitagorean theorem, Bessel inequality, Schwarz inequality,

polarization identity. Hilbert spaces, orthogonal complement, projection operators,

direct sum. Riesz theorem. Hilbert bases, countablity of the basis and separability,

Parseval identity. Lax-Milgram theorem. 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. Equicontinuity and Ascoli-Arzelà theorem.

4. Topology generated by a family of functions; weak topology in a

Banach space, basis of a weak topology; properties of weakly converging

sequences, weak and strong closure, Mazur lemma. Bidual space and

reflexive spaces; strong and weak continuity of linear operators. Weak*

topology, properties of weak* converging sequences, Banach-Alaoglu

theorem. Helly lemma, Kakutani theorem. Properties of reflexive (and

separable) Banach spaces. Sequential relative compactness theorem in a

reflexive Banach space. Weiertrass 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. Definition, 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.

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.

I meet students from Monday to Friday by email appointment.

- H. Brezis, Analisi funzionale, Liguori.

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

- K. Yosida, Functional analysis, Springer.

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

- W. Rudin, Analisi reale e complessa, Boringhieri.

- V. Checcucci, A. Vesentini, E. Tognoli, Lezioni di topologia generale, Feltrinelli.

- R. Wheeden, A. Zygmund, Measure and integral, CRC Press.