Banach spaces. Hahn-Banach Theorem and Applications. Hilbert spaces: the projection theorem and applications, complete orthonormal systems. Fixed points theorem. Duality, weak topologies. 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's 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
can be found at the link:
http://web.math.unifi.it/users/mascolo/DIDATTICA-MATEMATICA/libroAf.pdf
Learning Objectives
Knowledge acquired (at the end of the course):
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 to be used as requirements (required and/or recommended)
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
Historical development of functional analysis . Normed and Banach spaces . Dual spaces . Hahn-Banach Theorem and Applications . Separable and reflexive spaces . Problem of minimum norm and duality theory . Geometric form of the Hahn- Banach theorem. Linear Operators . Lemma Baire . Banach-Steihaus - theorem . Open mapping Theorem . Closed graph theorem . Fixed point Theorems in Banach spaces . Weak and weak * topology in normed spaces . Semicontinuous and convex functions . Properties of weak topologies and applications. Lp Spaces. Study of the weak topologies in the Lp spaces . Test functions and mollification. Fundamental lemma of the calculus of variations. Hilbert spaces . Projection theorem and applications. Bilinear forms and Stampacchia theorem . Lax- Milgram theorem and applications. Complete orthonormal systems. Sobolev spaces. Distribution and weak derivatives. Embedding and Compact Embedding Theorems. Variational Methods for differential equations. Regularity of solutions.. Calculus of Variations. Euler- Lagrange equation. Dirichlet's principle. Direct Methods. Application to the Minimum surface.
Bounded slope condition.