# Matematica | FUCNTION THEORY

## Matematica FUCNTION THEORY

 0512300023 DIPARTIMENTO DI MATEMATICA MATHEMATICS 2015/2016

 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
 ANTONIO VITOLO T PAOLA CAVALIERE
Objectives
1. KNOWLEDGE AND UNDERSTANDING: THE COURSE IS FOCUSED ON THE FUNDAMENTAL CONCEPTS OF MEASURE AND INTEGRATION THEORY IN RN TOGETHER WITH CALCULUS AND APPLICATIONS.
2. APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT WOULD BE FAMILIAR WITH THE METHODS TYPICAL OF MEASURE AND INEGRATION THEORY, AND WITH THE REDUCTION AND APPROXIMATION TECHNIQUE OF INTEGRAL CALCULUS.
3. MAKING JUDGEMENTS: THE STUDENT WILL BE ABLE TO COMPARE ALTERNATIVE PROOFS, TO UNDERSTAND THE ROLE OF THE ASSUMPTIONS AND TO CHOOSE THE MOST APPROPRIATE WAY TO OBTAIN A RESULT, LEARNING CRITICALLY.
4. COMMUNICATION SKILLS: THE STUDENT WILL BE ABLE TO COMMUNICATE AND EXPRESS WITH RIGOROUS AND EFFECTIVE EVIDENCE THE RESULTS OF THE LEARNED TOPICS AND TO PRESENT THEM WITH THE SUITABLE MOTIVATIONS, EXAMPLES AND COUNTEREXAMPLES.
5. LEARNING SKILLS: THE STUDENT WILL GET A MATHEMATICAL BACKGROUND WHICH ALLOWS HIM TO READ MORE ADVANCED TEXTS OF MATHEMATICAL ANALYSIS AND IN GENERAL OF SCIENTIFIC SUBJECTS WHICH USE MATHEMATICAL TOOLS.
Prerequisites
BASIC CONCEPTS ON FUNCTIONS ONE AND SEVERAL COMPLEX VARIABLE.
Contents
METRIC SPACES: DISTANCE AND TOPOLOGY. BOUNDEDNESS AND COMPLETENESS. COMPACTNESS. BANACH SPACES.
PEANO-JORDAN MEASURE AND RIEMANN INTEGRAL. LEBESGUE MEASURABLE FUNCTIONS. LEBESGUE MEASURE AND INTEGRATION. COMPARISON WITH PEANO-JORDAN MEASURE AND RIEMANN INTEGRAL.
THEOREMS ON TAKING LIMITS UNDER THE INTEGRAL SIGN. INTEGRATION TERM BY TERM. DERIVATION UNDER INTEGRAL SIGN.
MEASURE REDUCTION FORMULAS AND INTERCHANGING THE ORDER OF INTEGRATION: THEOREMS OF TONELLI AND FUBINI.
APPLICATIONS.
Teaching Methods
FRONTAL LECTURES.
Verification of learning
ORAL EXAM.
Texts
1. E. GIUSTI, ANALISI MATEMATICA 2, BOLLATI BORINGHIERI.
2. LECTURE NOTES.
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