COMPETENCIES

M08CM01: To know the basic concepts and techniques of Lebesgue Measure and Integration Theory.

M08CM02: To be able to relate the concept of measure with the concept of integration.

M08CM03: To know and employ the Theorems of Monotone and Dominated Convergence, Fatou's Lemma, Fubini's Theorem and the Theorem of Change of Variables.

M08CM04: To know the definition Banach and Hilbert spaces, and to be able to classify of the most useful typical examples in Functional Analysis, particularly, the spaces of sequences and of functions.

M08CM05: To use with precision the specific techniques of the theory of operators in normed spaces and Hilbert spaces.

M08CM07: To be able to develop rigorously the fundamental results of the theory.



LEARNING RESULTS

- To understand the fundamental concepts of Measure Theory and its application in the definition of the Lebesgue Integral.

- To apply the fundamental theorems of convergence to be able to recognize integrable functions.

- To know the basic examples of spaces of integrable functions and their metric properties.

- To know the fundamental properties of norm spaces and the linear transformations between them.

- To understand the concepts of scalar product and of Hilbert Space and their fundamental properties.

1. MEASURE OF SETS IN RN. MEASURE SPACES: The Riemann Integral and its limitations, content, exterior measure, Lebesgue measure, properties. No measurable sets, sigma-algebras, measures and measure spaces: basic properties and examples.



2. LEBESGUE INTEGRAL AND ITS PROPERTIES: integration of simple functions, measurable functions, integration of positive functions and of no definite sign, integrable functions, convergence theorems for integrals. Differentiation under the integral sign.



3. FUBINI'S THEOREM AND CHANGE OF VARIABLES: Integrals of functions of several variables, Tonelli's and Fubini's Theorems, change of variables.



4. INTRODUCTION TO HILBERT SPACES: scalar product, Cauchy-Schwartz inequality. Hilbert spaces. orthogonality and projections. Linear functionals: representation theorem. Orthogonal systems and bases.



5. INTRODUCTION TO BANACH AND LP SPACES: norm spaces, Lp spaces, Hölder and Minkowski inequalities. Completness of Lp. Linear operators: continuity and boundedness.



Problems and practical questions related to each lesson will be developed.

The theoretical contents will be presented in master classes following the basic bibliography. These classes will be complemented with problem classes and seminar sessions in which the students will solve proposed problems and will present complementary material related to their learning outcomes.



In the seminars, questions and examples representative of the content of the subject will be developed, which will generally have been provided to the student in advance to work on and will motivate subsequent reflection and discussion in the session dedicated to it. This process can be individual or in groups.



Moreover, depending on the characteristics of the group, ERAGIN type methods might implemented (see "ORIENTATIONS" below).

Written exam: Between 65% and 100% of the final mark. The student have to obtain a minimum of four points over ten in order to pass and in order that the other tasks are taking into consideration.



Evaluation of the tasks proposed and participation in the seminars: up to 35%.



In case of resignation to continuous evaluation, the evaluation will be given by the result of the final written exam, up to ten points.



ORIENTATIONS: in case of setting up ERAGIN type methods, the professor will explain the value that each task has in the final mark.

Virtual E-gela platform.

Basic bibliography

J. A. Facenda y F. J. Freniche, Integración de funciones de varias variables, Pirámide, Madrid, 2002.

A. García y Mª J. Muñoz Bouzo, Espacios de Hilbert y Análisis de Fourier: los primeros pasos, Ed. Sanz y Torres, Madrid, 2012.

M. De Guzman y R. Rubio, Integración: teoría y técnicas, Alhambra, Madrid, 1979.

R. Wheeden y A. Zygmund, Measure and integral, Marcel Dekker, 1977.

In-depth bibliography

H. Brezis, Análisis Funcional, Alianza, Madrid, 1984.
G. B. Folland, Real Analysis, John-Wiley-Interscience, New York, 1984.
H. L. Royden, Real Analysis, Macmillan, New York, 1963.
W. Rudin, Análisis real y complejo, Alhambra, Madrid, 1979.
T. Tao, An introduction to Measure Theory, American Mathematical Society, 2011.

Journals