| Area of Mathematics: | Analysis | ||
| Semester: | 4ο | ||
| Course ID: | 41101 | ||
| Course Type: | Compulsory | ||
| Teaching hours per week: | Theory: 4 | Practice: 1 | Laboratory: 0 |
| ECTS : | 6 | ||
| Eclass: | For the course’s material, click here. | ||
| Instructors: | Nikolaos Tsirivas | ||
Description
- Metric spaces: Definitions, basic properties and examples, topological concepts.
- Sequences in metric spaces, convergent sequences.
- Bounded and totally bounded sets. Compact metric spaces, examples.
- Characterization of compact metric spaces.
- Connectivity, connected components, examples.
- Complete metric spaces: Definition, basic properties, examples. Cantor and Baire theorems, applications.
- Equivalent metrics, relative metric, Separability.
- Limits of functions, continuity of functions in metric spaces, properties of continuous functions. Uniform continuity. Isometries, Lipschitz functions, Isomorphisms.
- Fixed point Theorem and applications in algebraic and differential equations.
- Cartesian products of metric spaces, examples, applications.
Bibliography
- Carothers, Real Analysis, Cambridge University Press, 2012
- H.Royden, P.Fitzpatrick, Real Analysis, 4th edition, Pearson Modern Classics for Advanced Mathematics Series, 2022.
