
Course unit
FUNCTIONAL ANALYSIS
SCP6076297, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2017/18
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/05 
Mathematical Analysis 
6.0 
Course unit organization
Period 
Second semester 
Year 
3rd Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
3.0 
24 
51.0 
No turn 
Lecture 
3.0 
24 
51.0 
No turn 
Prerequisites:

Classical topics in differential, integral, multivariable calculus, as well as basic notions of linear algebra and a few basic elements of measure and integration theory for which a lecture course in Real Analysis is recommended. 
Target skills and knowledge:

To get acquainted with the terminology, the fundamental notions and theorems of classical functional analysis in Banach and Hilbert space.
To acquire the ability to recognize the typical arguments of functional analysis in view of its possible applications. 
Examination methods:

Oral exam on all topics of the course, including all proofs of all propositions. The exam consists in answering questions aiming at estimating the knowledge gained by the student, and in discussing the notions and the presented results in order to estimate the level of familiarity of the student with the those notions, in particular by analyzing the details of the proofs of the theorems and the proposed examples and exercises. 
Assessment criteria:

Grades are decided starting from a first level (1823 out of 30) in which case the mere knowledge of all topics is required, passing to a second level (2427 out of 30) for which
familiarity with the studied notions is required, and finally
getting to a level of excellence (2830 out of 30) in which case critical thinking is required. 
Course unit contents:

The fundamental theorems of functional anlysis, HahnBanach Theorem, BanachSteinhaus Theorem, Open mapping and Closed graph Theorem. Weak and weak* topologies, reflexivity, separability, compactness. Applications to classical function spaces, L^p spaces and spaces of continuous functions in particular. Hilbert spaces, compact and selfadjoint operators, elements of spectral theory. 
Planned learning activities and teaching methods:

Traditional lectures with classical blackboard by using which it is possible to represent a large part of the proof of a theorem and analyze and discuss it in a critical way. 
Textbooks (and optional supplementary readings) 

Brezis, Haim, Functional analysis, sobolev spaces and partial differential equationsHaim Brezis.. New York: Springer, 2011.

Kolmogorov, Andrej Nikolaevič; Fomin, Sergej Vasilevic, Elementi di teoria delle funzioni e di analisi funzionaleAndrej N. Kolmogorov, Sergej V. Fomin. Roma: Editori riuniti, 2012.

Riesz, Frigyes; SzokefalviNagy, Bela; Boron, Leo F., Functional analysisFrigyes Riesz and Bela Sz. Nagytranslated from the 2. French edition by Leo F. Boron. New York: Dover, 1990.

Rudin, Walter, Functional analysisWalter Rudin. New York [etc.]: McGrawHill, .

Lax, Peter, Functional analysisPeter D. Lax. New York: Wiley, .

Innovative teaching methods: Teaching and learning strategies
 Interactive lecturing
 Story telling
 Problem solving
Innovative teaching methods: Software or applications used
Sustainable Development Goals (SDGs)

