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. Calculus of Variations: Euler-Lagrange Equation, Dirichlet principles, Direct Methods, Minimal Surfaces.
H. Brezis, Analisi Funzionale- Teoria ed Appl., Liguori Ed., Napoli,1986
Erwin Kreyszig, Introductory Functional Analysis with Applications, Ed. John Wiley & Sons, 1989
Elvira Mascolo, Appunti di Analisi Funzionale
reperibili all'indirizzo:
http://web.math.unifi.it/users/mascolo/DIDATTICA-MATEMATICA/libroAf.pdf
Learning Objectives
1 Knowledge acquired (at the end of the course):
2 Knowledge of the main theorems concerning the linear functional and operators. Weak Convergence and its applications to the study of the minimum of integral functional. Main theorems in Hilbert spaces. Sobolev spaces and applications to variational methods for the study of differential equations. Applications to Calculus of Variations.
Competence acquired:
Study of the theory of the Functional Analysis with an approach that takes account of its role as fundamental tool in solving the mathematical problems related to applied mathematics and physics.
Skills acquired (at the end of the course):
Solving abstract problems of normed, Banach and Hilbert spaces. Solving boundary value problems for differential equations with variational methods in Sobolev spaces.
Prerequisites
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
Further information
Attendance of lectures, practice and lab:
Not mandatory
Teaching tools:
http://web.math.unifi.it/users/mascolo/DIDATTICA-MATEMATICA/libroAf.pdf
Office hours:
Wednesday, 14:30-16 and the student's request.
Dipartimento di Matematica U.Dini
Viale Morgagni 67/a, Firenze
Contact:
Banach spaces. Hahn-Banach Theorem and Applications. Baire theorem, Banach-Steinhaus theorem and applications. Closed graph theorem. Hilbert spaces: the projection theorem, and Stampacchia theorem for the bilinear forms, theorem of Lax-Milgram, complete orthonormal systems. Duality, weak topologies. Main theorem and properties of the weak and weak* toplogies. Separability and reflexivity. Study of the weak topologies of the spaces Lp. Sobolev spaces: weak derivatives and main properties. Embedding Theorems. Variational Methods for the study of differential equations.Calculus of Variations: Euler-Lagrange Equation, Dirichlet principles, Direct Methods, Minimal Surfaces.