Linear Functional Analysis
Official Course Description
The main goal of the course is to give a presentation of the principles of functional analysis and their applications. These include the basic properties of Banach and Hilbert spaces as well as the spectral theory of bounded and compact linear operators.
The course treats fundamental properties of Banach and Hilbert spaces and the bounded linear operators defined on them:
- Banach spaces, the Hahn-Banach Theorem, weak convergence and weak precompactness of the unit ball.
- Hilbert spaces. Examples including L2 spaces. Orthogonality, orthogonal complement, closed subspaces, projection theorem. Riesz Representation Theorem.
- Orthonormal sets, Bessel's inequality. Complete orthonormal sets, Parseval's identity.
- Baire Category Theorem and its consequences for operators on Banach spaces (uniform boundedness, open mapping, inverse mapping and closed graph theorems). Strong convergence of sequences of operators.
- Bounded and compact linear operators on Banach spaces and their spectra.
- The spectral theorem for compact self-adjoint operators on Hilbert spaces.
For admission to the course, English 6 is required as well as at least 90 credits in pure mathematics including knowledge corresponding to a course in Integration Theory, 7.5 credits (Lebesgue integral).