# ADVANCED ANALYSIS - mod. A (529SM-1)

#### A.Y. 2020 / 2021

Professor
Period
First semester
Credits
6
Duration/Length
48
Type of Learning Activity
Programme specific subjects
Study Path
[PDS0-2018 - Ord. 2018] common
Mutuazione
Mutuato: SM34 - 529SM-1 - ADVANCED ANALYSIS - MOD. A
Syllabus
Teaching language

English

Learning objectives

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.

Prerequisites

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

Contents

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.

Teaching format

Lectures and solutions of problems.

Extended Programme

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.
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.

End-of-course test

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.

Other information

I meet students from Monday to Friday by email appointment.

Texts/Books

- 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.

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