


Politecnico di Torino  
Academic Year 2016/17  
02ACCMQ Functional analysis 

1st degree and Bachelorlevel of the Bologna process in Mathematics For Engineering  Torino 





Subject fundamentals
Functional Analysis can be considered as the extension of linear algebra to infinite dimensional vector spaces. All the integral and differential equations that arise either from 'pure' mathematics or from the applications can be seen as transformations (linear or nonlinear) between function spaces, and hence between infinite dimensional vector spaces. The aim of Functional Analysis is to provide a systematic treatment both for the study of function spaces and of transformations between them.

Expected learning outcomes
The knowledge obtained through this course consists in the fundamental mathematical tools for the understanding and the correct treatment of complex mathematical problems.
The skills consist in the ability to recognize the mathematical structures and instruments used in the solution of problems concerning integral or differential equazions and in the ability to solve at least the simplest cases. 
Prerequisites / Assumed knowledge
Mathematical Analysis I and II, Complex Analysis, Geometry,

Contents
Banach spaces and linear operators.
Hilbert spaces, projections, orthonormal basis. Generalized Fourier series. Dual spaces: linear functionals, weak convergence. Compactness in finite dimensional spaces. Compact operators and applications to integral equations. Fundamentals off spectral theories Distributions. Fourier transforms. Fundamentals of caluculus of Variations 
Texts, readings, handouts and other learning resources
A. Kolmogorov, S. Fomin, Elementi di teoria delle funzioni e di analisi funzionale, Mir (1980).
H. Brezis, Analyse fonctionnelle, Masson (1983) 
Assessment and grading criteria
Written exan with both theoretical and practical questions followed by an oral discussion

