Normed Vector Spaces: Basic denitions, continuity, convergence and compactness in normed spaces, separable spaces, Banach spaces, Banach fixed point theorem. Integration theory: Riemann integral, Lebesgue measure and Lebesgue integral, Lp-spaces. Linear Operators: Basic denitions, bounded linear operators, uniform boundedness theorem, Banach-Steinhaus theorem, open mapping theorem, closed graph theorem. Dual Spaces and Weak Topologies: Linear functionals and dual spaces, Hahn-Banach theorem, weak topology, reexive spaces, best approximation problem. Hilbert Spaces: Basic Denitions, Riesz representation theorem, orthogonality, orthogonal projection theorem, Gram-Schmidt orthonormalization, Fourier series in Hilbert spaces, dual Spaces and adjoint operators, normal, self-adjoint and unitary operators, Fourier transform. Partial differential equations: Distribution theory, Sobolev spaces, weak derivatives, Lax-Milgram theorem. Differential calculus in Banach spaces: Gateaux and Frechet derivatives. Spectral Theory for Bounded Linear Operators: Spectrum and resolvent, Neumann series. Linear system theory: Controllability, stability, observability of linear systems.
- Esame di Functional analysis in applied mathematics and engineering docente Prof. K. Engel
- Università: L'Aquila - Univaq
- CdL: Corso di laurea magistrale in ingegneria matematica