TOPICS IN ADVANCED ANALYSIS

Degree course: 
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree: 
2015/2016
Year: 
3
Academic year in which the course will be held: 
2017/2018
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
Second semester
Standard lectures hours: 
80
Detail of lecture’s hours: 
Lesson (80 hours)
Requirements: 

Knowledge of mathematical analysis and general topology as usually studied in a first level degree program in mathematics.

Final Examination: 
Orale

The exam is divided into two parts:
- a written part, consisting in solving two exercises chosen from a list of problems assigned at the end of the course, where the student shows to have acquired an operating knowledge of the subject and to be able to apply the techniques described in class,
- an oral examination consisting in the discussion of the exercises and to the proof of one or two theorems seen in class.

Assessment: 
Voto Finale

The aim of the course is to provide the fundamental notions of functional analysis, illustrating the methods and the most significant techniques of proof.

After the course students
- will know the statement and proof of the main theorems seen in class;
- will have acquired an operating knowledge of the methods and techniques of Functional Analysis;
- will be able to apply the abstract results studied in class to particular cases;
- will be able to apply the techniques of proof learned in class to prove theorems similar to thosee prove in class and to solve problems of theoretical nature.

Topological vector spaces, locally convex spaces. Metrizability and normability. Banach and Frechet spaces. The space of test functions, and of rapidly decreasing functions. The Hahn-Banach theorem and its consequences. The Baire, Banach-Steinhaus, open mapping and closed graph theorems.
Duality in Banach spaces, weak topologies, weak and strong convergence, double dual and reflexive spaces. Banach-Alaoglu theorem.
Uniform convexity and Milman Theorem. Eberlain-Schmulyan theorem. Extreme points and the Krein-Milman theorem. Introduction to spectral theory. Compact operators, Schauder theorem. Spectral theory of compact operator, Riesz-Shauder theory. Introduction to operator semigroups. The Hille-Yosida theorem.

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer
W. Rudin, Functional Analysis, Mc. Graw Hill
K. Yosida, Functional Analysis, Springer

Frontal lectures

Office hours: by appointment.

Borrowed from

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Degree course in: MATHEMATICS