# Matematica | FUCNTION THEORY

## Matematica FUCNTION THEORY

 0512300023 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2016/2017

 YEAR OF COURSE 3 YEAR OF DIDACTIC SYSTEM 2010 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 6 48 LESSONS SUPPLEMENTARY COMPULSORY SUBJECTS
 PAOLA CAVALIERE T
Objectives
KNOWLEDGE AND UNDERSTANDING
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, THAT IS CRUCIAL FOR ANY ADVANCED LEVEL COURSE IN ANALYSIS.
THE COURSE IS DESIGNED TO ENABLE STUDENTS TO
OINTERPRET THE MAIN CONCEPTS IN METRIC AND NORMED SPACES THEORY ANALYTICALLY, GRAPHICALLY AND VERBALLY;
ODEVELOP THEIR ABILITY TO THINK IN A CRITICAL MANNER,
OTO FORMULATE AND DEVELOP TREATED ARGUMENTS IN A LOGICAL MANNER;
OIMPROVE THEIR SKILLS IN ACQUIRING NEW UNDERSTANDING AND EXPERIENCE IN FUNCTIONAL ANALYSIS;
OANALYZE 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:
OKNOWLEDGE AND UNDERSTANDING
DEFINE AND STATE SOME OF THE BASIC DEFINITIONS OF CONCEPTS CONCERNING METRIC AND NORMED SPACES SUCH AS COMPACTNESS, CONNECTEDNESS, COMPLETENESS;
STATE SOME OF THE PRINCIPAL THEOREMS AS TREATED IN THE MODULE; USE SOME OF THE BASIC DEFINITIONS AND PRINCIPAL THEOREMS IN THE INVESTIGATION OF EXAMPLES;
PROVE BASIC PROPOSITIONS CONCERNING THOSE ASPECTS OF METRIC AND NORMED SPACES TREATED IN THE MODULE.
OINTELLECTUAL SKILLS
APPLY COMPLEX IDEAS TO FAMILIAR AND TO NOVEL SITUATIONS;
WORK WITH ABSTRACT CONCEPTS AND IN A CONTEXT OF GENERALITY;
REASON LOGICALLY AND WORK ANALYTICALLY;
PERFORM WITH HIGH LEVELS OF ACCURACY;
TRANSFER EXPERTISE BETWEEN DIFFERENT TOPICS IN MATHEMATICS.
OPROFESSIONAL SKILLS
SELECT AND APPLY APPROPRIATE METHODS AND TECHNIQUES TO SOLVE PROBLEMS;
JUSTIFY CONCLUSIONS USING MATHEMATICAL ARGUMENTS WITH APPROPRIATE RIGOUR;
COMMUNICATE RESULTS USING APPROPRIATE STYLES, CONVENTIONS AND TERMINOLOGY.
OTRANSFERABLE SKILLS
COMMUNICATE WITH CLARITY;
WORK EFFECTIVELY, INDEPENDENTLY AND UNDER DIRECTION;
ANALYSE AND SOLVE COMPLEX PROBLEMS ACCURATELY;
Prerequisites
THE PREREQUISITES ARE ONE AND SEVERAL VARIABLES CALCULUS, AND LINEAR ALGEBRA.
Contents
METRIC AND NORMED SPACES: DEFINITION OF METRIC SPACE AND EXAMPLES. NORMS AND INNER PRODUCTS ON VECTOR SPACES. METRIC NOTIONS: 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: CONTINUOUS MAPPINGS AND CONTINUOUS LINEAR OPERATOR. UNIFORMLY CONTINUOUS FUNCTIONS. LIPSCHITZ FUNCTIONS, ISOMETRIES AND CONTRACTIVE MAPS. HÖLDER FUNCTIONS. INFINITE DIMENSIONAL SPACES AND RELEVANT ISTANCES.

COMPLETE METRIC SPACES AND BANACH SPACE: COMPLETNESS AND CHARACTERIZATIONS. COMPLETATION OF A METRIC SPACE. EQUIVALENT METRICS. BAIRE’S THEOREM. BANACH FIXED POINT THEOREM, AND ITS APPLICATIONS.

CONNECTEDNESS AND COMPACTNESS IN METRIC SPACES; CONNECTED SETS, AND CONTINUOUS FUNCTIONS ON THEM. COMPACT AND SEQUENTIALLY COMPACT SETS. COMPACTNESS IN FINITE DIMENSIONAL SPACES. CRITERIA FOR COMPACTNESS IN METRIC SPACES. CONTINUITY AND COMPACTNESS. WEIESTRASS AND CANTOR THEOREMS. EXTENSION OF CONTINUOUS AND UNIFORMLY CONTINUOUS FUNCTIONS. COMPACTNESS CRITERIUM IN THE SPACE OF CONTINUOUS FUNCTIONS: EQUICONTINUITY AND ASCOLI-ARZELÀ THEOREM.
Teaching Methods
THE COURSE IS STRUCTURED AS A COMBINATION OF LECTURES AND PRATICAL SESSIONS. THE FEEDBACK FROM STUDENTS WILL ALWAYS BE HIGHLY APPRECIATED.
Verification of learning
THE EXAM CONSISTS IN AN ORAL EXAMINATION. STUDENT HAVE TO SHOW TO KNOW THE MAIN TOPICS AND THEIR CONNECTIONS.
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.