Content of the courses of Linear Algebra and Geometry, Calculus, Mathematical Analysis (especially: metric and normed spaces, sequences and series of functions) of the Undergraduate Degree in Mathematics.
To acquire knowledge and understanding on Function spaces, Measure theory and Lebesgue integration, Banach and Hilbert spaces.
1. Metric, normed and Banach spaces.
2. Measure and Integration.
3. Hilbert spaces.
1. Metric and normed spaces. Compactness. Completeness. Banach spaces.
Uniform convergence of sequences and series of functions. Spaces of bounded continuous functions. Continuous functions between metric spaces.
2. Measure and Integration: Measurable sets and functions. Measure spaces. Integration. Lebesgue’s Convergence Theorems. Lebesgue’s measure in R^n. L^p spaces. Holder’s and Minkowski’s inequalities. Norm. Completeness. Approximation with continuous functions.
3. Hilbert spaces: scalar product and orthogonality. Orthogonal projections onto closed subspaces. Representation of continuous linear functionals. Orthonormal systems. Bessel’s inequality. Riesz- Fischer Theorem. Orthonormal bases and Parseval’s identity. Convergence of trigonometric series.
W. Rudin: Principles of Mathematical Analysis, McGraw-Hill, 1976.
W. Rudin: Real and Complex Analysis, McGraw-Hill, 1987.
A. Kolmogorov, S. Fomin: Introductory Real Analysis, Dover, 1975.
Classroom lectures including exercises.
Written and oral examination. The written part consists mostly of exercises.
An attitude to the abstract side of mathematics will be helpful to the student.