Functional analysis

Here, you can dowload problem sets for the course Functional Analysis I gave at the University of Warsaw during the summer semester of 2018/19. For the lecture notes, see Lecture notes → Functional analysis.

Problem sets

• Part 1
  General properties of norms; distance from a subspace; hyperplanes. Linear functionals and continuity. Examples of Banach spaces.
• Part 2
  Basic properties of bounded linear operators. Minkowski functional. Dual norms and the Hahn-Banach theorem. Separation theorems and applications.
• Part 3
  The Riesz representation theorem and regular Borel measures. Properties of dual spaces; dual spaces in some particular cases. Weak convergence. Compact operators.
• Part 4
  Compact operators; spectrum; eigenvalues. Fredholm operators and the Fredholm index. Adjoint operators and their properties. The Riesz-Schauder theorem. The Fredholm alternative.
• Part 5
  Inner products and Hilbert spaces. The projection theorem and the Gram-Schmidt orthogonalization process. Orthonormal bases. Adjoint operators on Hilbert spaces; unitary operators. The Radon-Nikodym theorem.
• Part 6
  The Baire category theorem and its consequences: the Banach-Steinhaus theorem, the closed graph theorem and the open mapping theorem. Fourier series; completeness of the trigonometric system in \(L_2[0,2\pi]\). Fourier transform and convolution.
• Midterm test
• Solutions to midterm test
• Exam (1^st term)
• Exam (2^nd term)