Course guide of Functional Analysis (270113B)

Students are recommended to have successfully completed the basic and obligatory subjects of the first two years of the Degree in Mathematics.

• Normed spaces
• Hilbert spaces
• Compact linear operators on Hilbert spaces
• Duality in normed spaces
• Weak topologies

General goals:

• Abstraction capacity for the study of standard problems in Mathematical Analysis from the point of view of functional analysis, by understanding the advantages of functional analysis methods for solving various problems.
• Becoming familiar with some spaces of functions that are commondly employed in Mathematical Analysis and in its applications, like spaces of continuous (respectively, differentiable, analytic or harmonic, integrable, etc.) functions.
• Preparation for further studies, both in Mathematical Analysis and in other branches of Mathematics.

Specific goals:

• Employ the concepts of convergent sequence and Cauchy sequence in normed spaces, and understanding the notion of completeness.
• Being able to use the Hölder and Minkowski inequalities in concrete cases.
• Getting the skills to prove the continuity of some linear operators between normed spaces and compute their norms.
• Describe the dual space of some normed spaces.
• Being able to handle the orthogonal projection of a Hilbert space onto a closed subspace, and to compute the Fourier expansion with respect to the trigonometric system and Bessel's (in)equality.
• Study some examples of compact operators related to differential and integral equations.
• Compute the adjoint operator of some concrete operators on Hilbert spaces.
• Applications of the Hahn-Banach theorem.
• Check the reflexivity of some Banach spaces.
• Apply the Uniform boundedness principle.
• Formulate and understand the weak and weak-* topologies on some Banach spaces.

Chapter 1. Normed spaces.

Basic concepts and examples. Completeness. Banach's fixed point theorem. Continuous linear operators and functionals. Complemented subspaces. Quotients of normed spaces. Dual space of a normed space. Examples. Finite-dimensional normed spaces.

Chapter 2. Hilbert spaces.

Scalar Products and (pre-)Hilbert Spaces. Projection onto a closed convex set. Orthogonal projection theorem. Riesz-Fréchet representation theorem. Dual of a Hilbert space. Orthonormal bases. Operators on Hilbert spaces. Compact operators.

Chapter 3. Hahn-Banach theorem and Applications.

Analytical and geometric versions of the Hahn-Banach theorem. Separation of convex sets.

Chapter 4: Duality in normed spaces.

Duality for quotients and subspaces. Bidual of a normed space. Reflective spaces. The weak topology of a normed space and the weak-* topology of its dual space.

Chapter 5: The uniform boundeness principle and the closed graph theorem.

Baire category theorem. Banach-Steinhaus theorem. Applications. Open mapping theorem or Banach-Schauder theorem. Closed graph theorem. Applications.

The practices of this course will consist of workshops to solve exercises and problems related to the theoretical contents.

BREZIS, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.

CONWAY, J.K.: A Course in Functional Analysis, Springer-Verlag. New York, 1990.

MacCLUER, B.D.: Elementary Functional Analysis. Springer, 2009.

KADETS, V.; A Course in Functional Analysis and Measure Theory, Springer, 2018.

RINNE, P.R.; YOUNGSON,M.A.: Linear Functional Analysis. 2nd ed. Springer, 2008.

WILLEM, M.: Functional Analysis. Fundamentals and Applications. Birkhäuser, 2010.

BERBERIAN, S.K.: Lectures in Functional Analysis and Operator Theory.Springer-Verlag, New York, 1974.

DIEUDONNÉ,J.: History of Functional Analysis. North-Holland,Amsterdam,1981.

RUDIN, W.: Functional Analysis. McGraw-Hill, New York, 1973.

Lecture notes by Prof. Rafael Payá: https://www.ugr.es/~rpaya/cursosanteriores.htm (in Spanish)
Lecture notes by Prof. Javier Pérez: https://www.ugr.es/~fjperez/apuntes.html (in Spanish)

http://mathworld.wolfram.com/topics/FunctionalAnalysis.html

https://mathshistory.st-andrews.ac.uk/

https://encyclopediaofmath.org/wiki/Functional_analysis

• MD01. Lección magistral/expositiva 
• MD03. Resolución de problemas y estudio de casos prácticos 
• MD06. Análisis de fuentes y documentos 
• MD07. Realización de trabajos en grupo 
• MD08. Realización de trabajos individuales 

The evaluation will be conducted through a diversified evaluation system based on the following criteria:

- Attendance and active participation in theoretical and practical class sessions.
- Resolution of problems and proposed exercises.
- Participation in problem workshops - Written tests of a theoretical and practical nature.

The result of this diversified evaluation process will represent 30% of the final grade.

For the overall assessment of the knowledge assimilated and the skills acquired by the students, a final written test will be carried out, of a mandatory nature, which will consist of a practical part and a theoretical part. The score for this test will represent 70% of the final grade.

There will be a single final exam of a theoretical and practical nature, which will include all the contents of the subject. It must be done in person. The score obtained in this exam will represent 100% of the grade.

Students who, following the regulations of the UGR in the terms and deadlines required therein, take advantage of the "Single Final Assessment modality" for their evaluation, will take a single final exam that will consist of theory and problems covering all the contents of the subject. It must be done in person. The grade obtained in said exam will represent 100% of the final grade.