Real and Functional Analysis (9)
- Code: DT0626
- Unit Coordinator: Marco Di Francesco, Michele Palladino
- Programme: InterMaths
- ECTS Credits: 9
- Semester: 1
- Year: 1
- Campus: University of L'Aquila
- Language: English
- Aims:
Introducing basic tools of advanced real analysis such as metric spaces, Banach spaces, Hilbert spaces, bounded operators, weak convergences, compact operators, weak and strong compactness in metric spaces, spectral theory, in order to allow the student to formulate and solve linear ordinary differential equations partial differential equations, classical variational problems, and numerical approximation problems in an "abstract" form. Provide a primer of abstract measure and integration to be used in advanced probability and analysis courses.
- Content:
- Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
- Spaces of continuous functions. Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
- Crash course on abstract measure and integration. Measurable spaces and measurable functions. Borel and Lebesgue measures. Integrals on measure spaces. Limit exchange convergence theorems. Lp spaces. Product measures. Signed measures and Radon-Nicodym Theorem. Riesz representation theorem for measures.
- Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
- Hilbert spaces. Elementary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
- Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
- Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.
- Pre-requisites:
Basic calculus and analysis in several variables, linear algebra.
- Reading list:
- John K. Hunter, Bruno Nachtergaele, Applied Analysis. World Scientific.
- H. Brezis, Funtional Analysis, Sobolev Spaces, and partial differential equations. Springer.
- Piermarco Cannarsa, Teresa D’Aprile, Introduction to Measure Theory and Functional Analysis, Springer.
