Geometric Foundations of Analysis

The purpose of the course is to construct a natural bridge between basic courses such as Real Analysis, Real Multivariable Analysis and more advanced courses such as Topology, Differential Geometry, Calculus on Manifolds. We are going to introduce concepts of topological and metric spaces and to study main topological properties in that broad context.No prerequisites.


Curriculum:

  1. Introductory notions of topology: topological spaces, open and closed sets, continuous functions. Frequently used topological spaces.
  2. A rigorous definition of real numbers using Dedecind cuts. Complex numbers, Euclidean spaces, spaces of functions.
  3. Metric spaces. Converging sequences, Cauchy sequences in metric spaces. Complete metric spaces. Contraction mapping principle.
  4. Connected and path-connected topological and metric spaces. Intermediate value theorem.
  5. Compact sets in metric spaces, various equivalent definitions. Compactness, continuity and extremal properties.
  6. Mean value theorem. L’Hospital’s rule. Taylor’s rule.
  7. Sequences of functions. Uniform convergence, differentiation and integration. The Stone-Weierstrass theorem.
  8. Power series, Fourier series.
  9. Functions of several variables. The inverse function theorem, the implicit function theorem.

Textbooks

  • W.Rudin. Principles of mathematical analysis.
  • J.Munkres. Topology.