Topology and Linear Analysis

Recommended Prerequisites

Linear Algebra and Analytic Geometry I, II; Infinitesimal Analysis I, II, III.

Teaching Methods

The classes are essentially of expository nature and inculing  examples and exercises that lead the students to understanding and applying the material being taught.  The classes should be focused on the teaching of the reasoning processes, so that the students learn how to manipulate the objects presented along the course and more easily find out by themselves how to reach other results by independent reading or problem solving. Some tutorial support will be available to help the students solving the proposed tasks.

Learning Outcomes

Introduction of several abstract concepts from Topology and Analysis. Development of abstraction and argumentative abilities. Study of the most important properties of General Topology and its impact in Linear Analysis.

Work Placement(s)

No

Syllabus

I. Topology

Metric spaces versus continuous maps: definition and examples; open subsets.

Topological spaces: definition and examples; continuous maps, homeomorphisms; basis, neighbourhoods, interior, closure and border; subspaces and (finite) products; Hausdorff spaces.

Connected spaces: definition, examples and general properties; connected components; pathwise connected spaces.

Compact spaces: definition, examples and general properties. Compact metric  spaces.

Complete metric spaces: Cauchy sequences; complete spaces; compact spaces versus complete spaces; compactness in Rn.

 

II. Linear Analysis

Normed spaces and bounded linear operators: definition and examples; finite dimensional normed spaces; dual spaces; fundamental results (Hahn-Banach Theorem, Banach-Steinhaus Theorem, Open-Mapping Theorem and Closed-Graph Theorem).

Spaces with internal product: orthonormal sets; orthogonal projection; Bessel’s Inequality and Parseval’s Identities; Riesz Representation Theorem.

Head Lecturer(s)

João Filipe Cortez Rodrigues Queiró

Assessment Methods

Assessment
Exam(100%) or Test (75%) +Problem Resolving Report(25%): 100.0%

Bibliography

W.A. Sutherland, Introduction to Metric and Topological Spaces, Oxford, 2002.

 

B. Bollobás, Linear Analysis, An Introductory Course, Cambridge University Press, 1999