# Matematica | FUCNTION THEORY

## Matematica FUCNTION THEORY

 0512300023 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2022/2023

 YEAR OF COURSE 3 YEAR OF DIDACTIC SYSTEM 2018 SPRING SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 PAOLA CAVALIERE T
ExamDate
APPELLO TEORIA DELLE FUNZIONI14/02/2023 - 10:00
Objectives
THE EXPERIENCE GAINED FROM THE FIRST TWO YEARS OF ANALYSIS COURSES POINT TO THE IMPORTANCE OF PASSAGE TO THE LIMIT. THE BASIS OF THIS OPERATION IS THE NOTION OF DISTANCE BETWEEN ANY TWO POINTS OF THE LINE OR THE COMPLEX PLANE. THE ALGEBRAIC PROPERTIES OF UNDERLYING SETS OFTEN PLAY NO ROLE IN THE DEVELOPMENT OF ANALYSIS; THIS SITUATION NATURALLY LEADS TO THE STUDY OF METRIC SPACES. THE ABSTRACTION NOT ONLY SIMPLIFIES AND ELUCIDATES MATHEMATICAL IDEAS THAT RECUR IN DIFFERENT GUISES, BUT ALSO HELPS ECONOMIZE THE INTELLECTUAL EFFORT INVOLVED IN LEARNING THEM. THE LANGUAGE IN WHICH A LARGE BODY OF IDEAS AND RESULTS OF FUNCTIONAL ANALYSIS ARE EXPRESSED IS THAT OF METRIC SPACES. THUS THE COURSE ATTEMPTS TO PROVIDE A LEISURELY APPROACH TO THE THEORY OF METRIC SPACES. THE PROOF SKILLS AND MATHEMATICAL KNOWLEDGE DEVELOPED IN THIS CLASS WILL PREPARE STUDENTS FOR ADVANCED MATHEMATICAL COURSES AND WILL BENEFIT THEM IN VARIOUS AREAS OF PURE AND APPLIED MATHEMATICS, SUCH AS TOPOLOGY, GEOMETRY, DIFFERENTIAL EQUATIONS, SCIENTIFIC COMPUTING, AND OPTIMIZATION.

KNOWLEDGE AND UNDERSTANDING

THE COURSE IS DESIGNED TO ENABLE STUDENTS TO
1. INTERPRET THE MAIN CONCEPTS IN METRIC AND NORMED SPACES THEORY ANALYTICALLY, GRAPHICALLY AND VERBALLY; 2. DEVELOP THEIR ABILITY TO THINK IN A CRITICAL MANNER; 3. FORMULATE AND DEVELOP TREATED ARGUMENTS IN A LOGICAL MANNER; 4. IMPROVE THEIR SKILLS IN ACQUIRING NEW UNDERSTANDING AND EXPERIENCE IN FUNCTIONAL ANALYSIS; 5. ANALYZE MATHEMATICAL PROBLEMS IN THE FRAMEWORK OF FUNCTION SPACES AND SOLVE IT USING A WIDE ARRAY OF TOOLS.

APPLYING KNOWLEDGE AND UNDERSTANDING

A STUDENT WHO COMPLETES THIS MODULE SUCCESSFULLY SHOULD BE ABLE TO

- KNOWLEDGE AND UNDERSTANDING:
1. DEFINE AND STATE SOME OF THE BASIC DEFINITIONS OF CONCEPTS CONCERNING METRIC AND NORMED SPACES;
2. STATE SOME OF THE PRINCIPAL THEOREMS AS TREATED IN THE MODULE;
3. USE SOME OF THE BASIC DEFINITIONS AND PRINCIPAL THEOREMS IN THE INVESTIGATION OF EXAMPLES;
4. PROVE BASIC PROPOSITIONS CONCERNING THOSE ASPECTS OF METRIC AND NORMED SPACES TREATED IN THE MODULE.

- INTELLECTUAL SKILLS:
1. APPLY COMPLEX IDEAS TO FAMILIAR AND TO NOVEL SITUATIONS;
2. WORK WITH ABSTRACT CONCEPTS AND IN A CONTEXT OF GENERALITY;
3. REASON LOGICALLY AND WORK ANALYTICALLY;
4. PERFORM WITH HIGH LEVELS OF ACCURACY;
5. TRANSFER EXPERTISE BETWEEN DIFFERENT TOPICS IN MATHEMATICS.

- PROFESSIONAL SKILLS:
1. SELECT AND APPLY APPROPRIATE METHODS AND TECHNIQUES TO SOLVE PROBLEMS;
2. JUSTIFY CONCLUSIONS USING MATHEMATICAL ARGUMENTS WITH APPROPRIATE RIGOUR;
3. COMMUNICATE RESULTS USING APPROPRIATE STYLES, CONVENTIONS AND TERMINOLOGY;
4. COMMUNICATE WITH CLARITY;
5. WORK EFFECTIVELY, INDEPENDENTLY AND UNDER DIRECTION;
6. ANALYSE AND SOLVE COMPLEX PROBLEMS ACCURATELY;
7. ADOPT EFFECTIVE STRATEGIES FOR STUDY.
Prerequisites
THE PREREQUISITES ARE ONE AND SEVERAL VARIABLES CALCULUS, AND LINEAR ALGEBRA.
Contents
METRIC AND NORMED SPACES (12 HOURS): DEFINITION OF METRIC SPACE AND EXAMPLES. NORMS AND INNER PRODUCTS ON VECTOR SPACES. SOME FUNDAMENTAL INEQUALITIES. BASIC NOTIONS IN METRIC SETTING: DISTANCE FROM A SET, BALLS, DIAMETER OF A SET, BOUNDED AND TOTALLY BOUNDED SET, CAUCHY AND CONVERGENT SEQUENCES; OPEN AND CLOSED SETS, INTERIOR, CLOSURE AND BOUNDARY OF A SET, DENSE SETS. SEPARABLE SPACES. FINITE DIMENSIONAL SPACES.

CONTINUOUS MAPPINGS (8 HOURS): CONTINUOUS MAPPINGS. TOPOLOGICALLY EQUIVALENT METRICS. ISOMETRIES. BOUNDED LINEAR OPERATOR AND THEIR CHARACTERIZATIONS. UNIFORMLY CONTINUOUS FUNCTIONS. LIPSCHITZ FUNCTIONS AND CONTRACTIVE MAPS. HÖLDER FUNCTIONS. LIPSCHITZ EQUIVALENT METRICS. INFINITE DIMENSIONAL SPACES AND RELEVANT ISTANCES.

COMPLETE METRIC SPACES AND BANACH SPACE (10 HOURS): COMPLETNESS AND CHARACTERIZATIONS. COMPLETATION OF A METRIC SPACE AND OF A NORM SPACE. RELEVANT ISTANCES OF COMPLETE AND UNCOMPLETE SPACES. EQUIVALENT NORMS. BANACH-CACCIOPPOLI FIXED POINT THEOREM AND SEVERAL APPLICATIONS.

COMPACTNESS IN METRIC SPACES (14 HOURS): COMPACT AND SEQUENTIALLY COMPACT SETS. COMPACTNESS IN FINITE DIMENSIONAL SPACES. CRITERIA FOR COMPACTNESS IN METRIC SPACES AND IN NORMED SPACES. CONTINUITY AND COMPACTNESS. WEIESTRASS AND CANTOR THEOREMS. EXTENSION OF CONTINUOUS AND UNIFORMLY CONTINUOUS FUNCTIONS.

COMPACTNESS CRITERIUM IN THE SPACE OF CONTINUOUS FUNCTIONS (4 HOURS): EQUICONTINUITY AND ASCOLI-ARZELÀ THEOREM. APPLICATIONS.
Teaching Methods
THE COURSE IS STRUCTURED AS A COMBINATION OF LECTURES AND PRATICAL SESSIONS. THE LAST ONES WILL SHOW THAT THE ABSTRACTION OF METRIC SPACES SETTING NOT ONLY SIMPLIFIES AND ELUCIDATES MATHEMATICAL IDEAS THAT RECUR IN DIFFERENT GUISES, BUT ALSO HELPS ECONOMIZE THE INTELLECTUAL EFFORT INVOLVED IN LEARNING THEM. THE FEEDBACK FROM STUDENTS WILL ALWAYS BE HIGHLY APPRECIATED.
Verification of learning
THE EXAM CONSISTS IN AN ORAL OR WRITTEN EXAMINATION, THE STUDENT SHOULD EXPRESS HIS PREFERENCE. DURING THE WRITTEN EXAMINATION, STUDENTS ARE NOT ALLOWED TO USE TEXTBOOKS, NOTES, SCIENTIFIC CALCULATORS OR OTHER ELECTRONIC DEVICES. STUDENTS WILL BE EVALUATED ON THEIR ABILITY TO DEVISE, ORGANIZE AND PRESENT THE MAIN TOPICS AND THEIR CONNECTIONS.
THE MINIMUM GRADE OF 18/30 IS OBTAINED WHEN THE STUDENT EXHIBITS A SUFFICIENT KNOWLEDGE OF ALL THE PROGRAM, WITH THE EXCEPTION OF PROOFS.
THE MAXIMUM GRADE OF 30/30 WHEN THE STUDENT EXHIBITS A DEEP AND EXTENDED KNOWLEDGE OF THE THEORY AND THE APPLICATIONS OF ALL THE ARGUMENTS OF THE COURSE.
THE MAXIMUM GRADE CUM LAUDE REQUIRES A KNOWLEDGE OF ALL THE ARGUMENTS WITHIN CONTEXTS GOING BEYOND THE ONES INVESTIGATED IN THE COURSE.
Texts
S.K. BERBERIAN: FUNDAMENTALS OF REAL ANALYSIS, SPRINGER-VERLAG, 1999, 479 PAGES, ISBN: 0-387-98480-1
A.N. KOLMOGOROV - S.V. FOMIN: INTRODUCTORY REAL ANALYSIS, COURIER CORPORATION, 2012, 416 PAGES, ISBN 0-486-13474-1,
W. RUDIN: PRINCIPLES OF MATHEMATICAL ANALYSIS, THIRD EDITION, MCGRAW-HILL, SINGAPORE, 1976, 351 PAGES, ISBN: 0-07-054235-X

W.A. SUTHERLAND: INTRODUCTION TO METRIC AND TOPOLOGICAL SPACES, OXFORD, NEWYORK, 1981, 181 PAGES, ISBN: 0-19-853161-3

THE STUDENT CAN USE ANY GOOD TEXT THAT CONTAINS THE ARGUMENTS OF THE PROGRAM, SINCE IT IS A STANDARD PROGRAM. STUDENTS ARE URGED TO CHECK IN ADVANCE WITH THE TEACHER THE APPROPRIATENESS OF THE CHOSEN TEXT. HOWEVER, FURTHER DIDACTIC MATERIAL WILL BE AVAILABLE FOR ATTENDING STUDENTS.