LEARNING OUTCOMES

Introduction of the fundamental concepts of Lebesgue's Integration Theory and of Functional Analysis.

AIMS AND LEARNING OUTCOMES

Aims

The aim of this course is to teach some classical topics in Mathematical Analysis (Functional Analysis and Measure Theory), which are considered fundamental for a basic knowledge of Mathematics and for the students who plan to continue their studies with a Master's degree in Mathematics.

Expected learning outcomes

At the end of the course, the student will have to know the theoretical concepts introduced in the lectures, construct and discuss examples related to each of them (in such a way to better understand the abstract concepts), write/reconstruct the proofs seen in the lectures or easy variants of those and solve problems on the topics of the course.

PREREQUISITES

Mathematical Analysis I, 2, 3 and 4, Linear Algebra and Analitic Geometry, Geometry 1.

TEACHING METHODS

The course consists of frontal lectures carried out by the teacher where the theory is explained and where basic examples are discussed (four hours per week). These are integrated with problem lectures (one hour per week). The teaching material, including problem sheets and old exam scripts, is available in aulaweb.

SYLLABUS/CONTENT

• Normed and Banach Spaces.
• Hahn-Banach theorem, Banach-Steinhaus theorem, open mapping and closed graph theorems.
• Complements of Hilbert spaces.
• L^p spaces: Hölder and Minkowsky inequalities, Riesz-Fischer theorem, density properties.
• Convergences of measurable functions: convergence in measure, almost uniform convergence, Severini-Egoroff theorem.
• Radon-Nikodym theorem.
• Dual of L^p.

RECOMMENDED READING/BIBLIOGRAPHY

• M. Reed, B. Simon - Functional Analysis - Academic Press 1981
• B. Simon - Real Analysis, A Comprehensive Course in Analysis, Part 1 - AMS 2015
• H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer 2011
• N. Dunford, J.T. Schwartz - Linear Operators. Part I: General Theory - Interscience 1957
• W. Rudin - Real and complex analysis - McGraw-Hill Education
• A.E. Taylor, D.C. Lay - Introduction to Functional Analysis - Wiley and Sons 1980
• C.M. Marle - Mesures et Probabilités - Hermann 1974