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B018754 - FUNCTIONAL ANALYSIS
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Further information
Type of Assessment
Course program
Academic Year 2016-17
Coorte 2016 - Second Cycle Degree in MATHEMATICS
Course year
First year -
Belonging Department
Mathematics and Computer Science "Ulisse Dini" (DIMAI)
Course Type
Single education field course
Scientific Area
MAT/05 - MATHEMATICAL ANALYSIS
Credits
9
Teaching Hours
72
Teaching Term
⇒
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Mutuality
Course teached as:
B018754 - ANALISI FUNZIONALE
Second Cycle Degree in MATHEMATICS
Curriculum GENERALE
B018754 - ANALISI FUNZIONALE
Second Cycle Degree in MATHEMATICS
Curriculum GENERALE
Teaching Language
Italian
Course Content
Banach spaces. Hahn-Banach Theorem and Applications. Hilbert spaces: the projection theorem and applications, complete orthonormal systems. Duality, weak topologies. Separability and reflexivity. Study of the weak topologies of the spaces Lp. Sobolev spaces: weak derivatives and main properties. Embedding Theorem. Variational Methods for the study of differential equations. Direct Methods of the Calculus of Variations. Fixed point theorems.
Suggested readings (Search our library's catalogue)
-H. Brezis, Analisi Funzionale- Teoria ed Appl., Liguori Ed., Napoli,1986
-Erwin Kreyszig, Introductory Functional Analysis with Applications, Ed. John Wiley & Sons, 1989
L. C. Evans, Partial Differential Equations, AMS, Vol.19
-Elvira Mascolo, Appunti di Analisi Funzionale
at web site: http://web.math.unifi.it/users/mascolo/DIDATTICA-MATEMATICA/libroAf.pdf
-Erwin Kreyszig, Introductory Functional Analysis with Applications, Ed. John Wiley & Sons, 1989
L. C. Evans, Partial Differential Equations, AMS, Vol.19
-Elvira Mascolo, Appunti di Analisi Funzionale
at web site: http://web.math.unifi.it/users/mascolo/DIDATTICA-MATEMATICA/libroAf.pdf
Learning Objectives
Knowledge of the main theorems concerning the linear functional and operators. Weak Convergence and its applications to the study of he minimizer of integral functional. Main theorems in Hilbert spaces. Sobolev spaces and applications to variational methods for the study of differential equations. Applications of the fixed point results to the partial differential equations.
Prerequisites
Courses to be used as requirements (required and/or recommended)
Courses required: Mathematical Analysis III
Courses recommended: Fundamentals of Higher Analysis)
Courses required: Mathematical Analysis III
Courses recommended: Fundamentals of Higher Analysis)
Teaching Methods
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 225
Hours reserved to private study and other individual formative activities:153
Hours for lectures: 72
Hours reserved to private study and other individual formative activities:153
Hours for lectures: 72
Further information
Attendance of lectures, practice and lab:
Not mandatory
Not mandatory
Type of Assessment
The final exam is designed to ensure the acquisition of knowledge and ability through the solution of some exercises.
The oral test consists of a technical conversation with the teacher to bring out the ability capacity to solve problems and to check the ability of the student to explain in clear and correct way the arguments of a assigned theorems.
The oral test consists of a technical conversation with the teacher to bring out the ability capacity to solve problems and to check the ability of the student to explain in clear and correct way the arguments of a assigned theorems.
Course program
Hahn-Banach theorem and applications. Hilbert spaces: the projection theorem, duality, notes on orthonormal systems. Banach spaces: theory and notable examples. Duality, weak topologies. Separability and reflexivity. Sobolev spaces in one dimension: weak derivative and main properties. Variational formulation of ordinary differential equations. Sobolev spaces in multiple dimensions. Variational formulation of partial differential equations . Direct methods of the calculus of variations. Theorems of fixed point and Applications.