Description
Objectives
To master concepts and techniques in Hilbert and Banach spaces, and in the theory of linear operators in Hilbert and Banach spaces.
Syllabus
Bounded linear operators in Hilbert spaces: linear functionals, adjoint, self- adjoint, unitary and normal operators; projections; invariant subspaces; compact operators. Spectral theory of compact self-adjoint operators; spectra and numerical range of bounded operators; the spectral theorem; operational calculus for compact self-adjoint operators. Banach Spaces; quotient spaces, linear functionals and dual spaces; the Hahn-Banach theorem; reflexive spaces; the open mapping and closed graph theorems; the uniform boundedness principle. Banach Algebras; definition and examples; invertibility; ideals, maximal ideals and the quotient algebra; spectra. Topics: Compact operators in Banach spaces; Fredholm operators: Unbounded operators in Hilbert spaces.
Prerequisites
Basic knowledge of topology and measure theory.
Cross Competence Component
The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.
Ethical Principles
All members of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.