Università degli Studi di Urbino Carlo Bo / Portale Web di Ateneo


ELEMENTS OF FUNCTIONAL ANALYSIS AND NUMERICAL METHODS
ELEMENTI DI ANALISI FUNZIONALE E METODI NUMERICI

A.Y. Credits
2021/2022 9
Lecturer Email Office hours for students
Giovanni Molica Bisci
Teaching in foreign languages
Course with optional materials in a foreign language English
This course is entirely taught in Italian. Study materials can be provided in the foreign language and the final exam can be taken in the foreign language.

Assigned to the Degree Course

Applied Informatics (LM-18)
Curriculum: PERCORSO COMUNE
Date Time Classroom / Location
Date Time Classroom / Location

Learning Objectives

Aim of the course is to give students some basic tools and topics in Functional Analysis and Numerical Methods.

Program

01. Normed spaces, metric spaces, topology
01.01 Normed vector spaces
01.02 Euclidean spaces: inner product
01.03 Metric spaces
01.04 Topological spaces
01.05 Limits
01.06 Product of topological spaces
01.07 Linear functions between vectorial spaces
01.08 Norms. Equivalent norms
01.09 Continuity of multilinear functions
01.10 Sequences and subsequences
01.11 Metric spaces: completeness
01.12 Normed spaces and completeness: canonical examples
01.13 Equivalent norms in finite vectorial spaces
01.14 Sequential notion of compactness
01.15 Subspaces of compact spaces
01.16 Continuous functions on compact spaces
01.17 Product of compact spaces
01.18 Applications
01.19 Connected spaces
01.20 The Banach-Caccioppoli theorem

02 Integrable Riemann functions
02.01 Step functions
02.02 Integral of step function with compact support 
02.03 Measure of a set in the plane
02.04 Integrable Riemann functions
02.05 Properties of the integrals
02.06 Geometrical meaning of the integral
02.07 Observations on the integral concept
02.08 Extended notion in an interval
02.09 Local integrability of continuous functions
02.10 Extended Integral over an oriented interval
02.11 Indefinite integrals
02.12 Dedeking property 

03 Sets
03.1 The Cantor world
03.2 Relations
03.4 Preorders; Equivalence; Orders
03.5 Functions between sets
03.6 Image and preimage
03.7 Cartesian product between sets
03.8 Elements of GBN Theory
03.9 The choice axiom: equivaent versions
03.10 Cardinality (n the Frege sense): generality
03.11 Fundamentals theorems between cardinals and combinatorics

Bridging Courses

Some basic knowledges of Mathematical Analysis and Linear Algebra are necessary.

Learning Achievements (Dublin Descriptors)

Knowledge and understanding. At the end of the course the student must have acquired a good knowledge of the mathematical topics covered in the course. He must be able to argue correctly and with language properties on the topics covered in the program. Examples and working methods are shown in the classroom during the lessons and proposed in the exercises.

Applied knowledge and understanding. At the end of the course the student must have acquired a good ability to use the main tools of basic mathematics. He must be able to correctly apply the formulation studied and must be able to solve general mathematical problems similar to those studied. In particular, he must be able to apply the acquired knowledge even in contexts slightly different from those studied, and have the ability to use the acquired knowledge to independently solve problems that may appear new. Examples of such applications are shown in the classroom during the lessons and proposed in the exercises.

Autonomy of judgment. At the end of the course the student must have acquired a good ability to analyze topics and problems in general mathematics, the ability to critically evaluate any proposed solutions, and a correct interpretation of similar topics.

Communication skills. At the end of the course the student must have acquired a good ability to clearly communicate his / her statements and considerations concerning general mathematics problems. The working method is shown in the classroom during the lessons and proposed in the exercises.

Ability to learn. At the end of the course the student must have acquired a good capacity for autonomy in the study of the discipline, in the reading and interpretation of a qualitative phenomenon, in the search for useful information to deepen the knowledge of the topics covered.

Teaching Material

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Supporting Activities

There are no supporting activities.


Didactics, Attendance, Course Books and Assessment

Didactics

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

H. Brézis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011, xiv+599 pp.

G. De Marco, Analisi Due. Teoria ed Esercizi. Zanichelli, Bologna, 1999.

T. Jech, Set Theory. The Third Millennium Edition, revised and expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xiv+769 pp.

W. Rudin. Principles of Mathematical Analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill, New York-Auckland-Düsseldorf, 1976.

S. Salsa, Partial Differential Equations in Action, From Modelling to Theory  - Springer, 2007. 

Assessment

The expected objectives are verified through the following two tests, both mandatory:

1. A formative assessment test: consisting of a written paper - lasting 2 hours and 30 minutes - divided into five exercises on the following topics:

  • A theme
  • Two exercises
  • on the treated theoretical arguments.
    2. An oral interview: consisting of the discussion of the written paper and three open questions on the theoretical topics covered in the course.

    For both tests, the evaluation criteria are as follows:
    - relevance and effectiveness of the responses in relation to the contents of the program;
    - the level of articulation of the response;
    - adequacy of the disciplinary language used.

    The overall evaluation is expressed with a mark out of thirty taking into account both tests.

    Additional Information for Non-Attending Students

    Didactics

    Theorical and practical lessons.

    Attendance

    Although strongly recommended, course attendance is not mandatory.

    Course books

    H. Brézis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011, xiv+599 pp.

    G. De Marco, Analisi Due. Teoria ed Esercizi. Zanichelli, Bologna, 1999.

    T. Jech, Set Theory. The Third Millennium Edition, revised and expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xiv+769 pp.

    W. Rudin. Principles of Mathematical Analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill, New York-Auckland-Düsseldorf, 1976.

    Assessment

    The expected objectives are verified through the following two tests, both mandatory:

    1. A formative assessment test: consisting of a written paper - lasting 2 hours and 30 minutes - divided into five exercises on the following topics:

  • A theme
  • Two exercises
  • on the treated theoretical arguments.
    2. An oral interview: consisting of the discussion of the written paper and three open questions on the theoretical topics covered in the course.

    For both tests, the evaluation criteria are as follows:
    - relevance and effectiveness of the responses in relation to the contents of the program;
    - the level of articulation of the response;
    - adequacy of the disciplinary language used.

    The overall evaluation is expressed with a mark out of thirty taking into account both tests.

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