To acquire a good knowledge of the theory of abstract integration. Introduction to functional analysis: Banach and Hilbert spaces.
Curriculum
teacher profile teaching materials
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.
Fruizione: 20402085 AM310 - ISTITUZIONI DI ANALISI SUPERIORE in Matematica L-35 N0 ESPOSITO PIERPAOLO
Programme
1. Abstract integration theoryRiemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.
Core Documentation
"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.Type of delivery of the course
The course plans lectures and exercises. Attendance is not required but strongly suggested.Type of evaluation
The written exam lasts three hours and evaluates the student's ability in solving exercises even of theoretical nature. The oral exam has a variable length and verifies the understanding of the topics developed during the lectures. The student might be exempted by the written exam if he passes a written intermediate test on the first part of the course and a final one on the second part of the course, each of three hours. teacher profile teaching materials
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.
Fruizione: 20402085 AM310 - ISTITUZIONI DI ANALISI SUPERIORE in Matematica L-35 N0 ESPOSITO PIERPAOLO
Programme
1. Abstract integration theoryRiemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.
Core Documentation
"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.Type of delivery of the course
The course plans lectures and exercises. Attendance is not required but strongly suggested.Type of evaluation
The written exam lasts three hours and evaluates the student's ability in solving exercises even of theoretical nature. The oral exam has a variable length and verifies the understanding of the topics developed during the lectures. The student might be exempted by the written exam if he passes a written intermediate test on the first part of the course and a final one on the second part of the course, each of three hours.