Course guide of Mathematical Methods 3 (2671126)

• Linear Algebra and Geometry I and II, Calculus I, and Mathematical Methods I
• When utilizing AI tools for this course, students are expected to adopt an ethical and responsible approach. The recommendations outlined in the "Recommendations for the Use of Artificial Intelligence at the UGR" document, available at this link: https://ceprud.ugr.es/formacion-tic/inteligencia-artificial/recomendaciones-ia#contenido0, must be followed.

• Hilbert spaces
• Series expansions, eigenfunctions

That the student understands the general concepts of Hilbert spaces, especially regarding their application to Physics, and is capable of solving the associated problems.

Unit 1. Normed spaces and Banach spaces.

Unit 2. Euclidean spaces and Hilbert spaces.

Unit 3. Function spaces and series expansions.

Unit 4. Functionals and distributions.

Unit 5. Linear operators.

Unit 6. Introduction to spectral theory

1. L. Abellanas y A. Galindo, Espacios de Hilbert, Eudema, 1987.

2. S. K. Berberian, An Introduction to Hilbert Space. American Mathematical Society. 2nd edition (1999).

3. P. García González, J. E. Alvarellos Bermejo y J. J. García Sanz, Introducción al formalismo de la mecánica cuántica, U.N.E.D., Madrid, 2001.

4. G. Helmberg, Introduction to spectral theory in Hilbert space, North Holland, 1969.

5. R. P. Kanwall, Generalized functions (theory and technique), Academic Press, 1983.

6. A. N. Kolmogórov y S.V. Fomín, Elementos de la teoría de funciones y del análisis funcional, M.I.R., 1975.

7. N. Young, An Introduction to Hilbert Space, Cambridge University Press, 2012.

8. J.I. Tello del Castillo, Introducción al análisis funcional, Sanz y Torres, 2024.

9. R.D. Richtmyer, Principles of Advanced Mathematical Physics, vol. 1, Springer-Verlag, 1978.

10. P. Roman, Some modern mathematics for physicists and other outsiders, vol. 2, Pergamon, 1975.

11. A. Vera López y P. Alegría Ezquerra, Un curso de Análisis Funcional. Teoría y problemas, AVL, 1997.

12. E. Romera Gutiérrez, M. C. Boscá Díaz-Pintado, F. Arias de Saavedra Alías, F. J. Gálvez Cifuentes, J. I. Porras Sánchez, Métodos Matemáticos: Problemas de Espacios de Hilbert, Operadores lineales y Espectros, Paraninfo, 2013.

• MD01. Theoretical classes

Assessment will be based primarily on exams; additionally, the completion of proposed problem sets and/or assignments to be solved individually will be considered, through which students must demonstrate their acquired knowledge and understanding of the material.

• In the ordinary call (regular exam session), the final exam grade will account for 70% of the final mark (A), and the remaining 30% (B) will be evaluated complementarily based on one or more of the following criteria: submission of problem sets, written tests, and/or quizzes during class hours.
• To pass the course, it is an essential requirement to obtain a score of at least 4 out of 10 on the final exam.
• Assessment due to special circumstances (Evaluación por incidencias): Students who are unable to attend the final assessment exams (ordinary, extraordinary, or single final calls) or any officially scheduled assessments listed in the Course Syllabus (Guía Docente) due to any of the circumstances outlined in Article 9 of the Assessment and Grading Regulations for Students of the University of Granada, may request an alternative assessment following the procedure indicated in the aforementioned regulations.

• Final exam with theoretical questions and problems related to the material taught in class.
• As a general rule, the exam will account for 100% of the grade; however, at the student's request, the exam will be weighted at 70%, with the remaining 30% coming from the grade obtained in section B of the ordinary assessment. In the latter case, to pass the course, it is an essential requirement to obtain a score of at least 4 out of 10 on the final exam.

• In accordance with the Assessment and Grading Regulations for Students of the UGR, provision is made for a single final assessment (evaluación única final). This may be chosen by students who cannot comply with the continuous assessment method due to any of the reasons outlined in Article 8. To apply for the single final assessment, the student must submit a request through the electronic office (sede electrónica) during the first two weeks of the course, within the two weeks following their enrollment if it occurred later, or at a later date if unexpected circumstances arise, stating and providing evidence for the reasons preventing them from following the continuous assessment system. The exam will account for 100% of the grade.
• The single final assessment will consist of an exam featuring theory and/or problem.

Students with Specific Educational Support Needs (NEAE).

Following the recommendations of the CRUE and the Secretariat for Inclusion and Diversity of the UGR, the competency acquisition and assessment systems included in this course syllabus will be applied in accordance with the principle of universal design (design for all), facilitating learning and the demonstration of knowledge according to the needs and functional diversity of the student body. Teaching methodology and assessment will be adapted for students with NEAE, in accordance with Article 11 of the Assessment and Grading Regulations for Students of the UGR, published in the Official Bulletin of the UGR No. 112, on November 9, 2016. Inclusion and Diversity of the UGR. In the case of students with disabilities or other NEAE, the tutorial system must be adapted to their needs, in accordance with the recommendations of the UGR Inclusion Unit, with Departments and Centers establishing the appropriate measures to ensure that tutorials take place in accessible locations. Likewise, at the request of the teaching staff, support may be requested from the competent unit of the University when dealing with special methodological adaptations.

Información de interés para estudiantado con discapacidad y/o Necesidades Específicas de Apoyo Educativo (NEAE): Gestión de servicios y apoyos (https://ve.ugr.es/servicios/atencion-social/estudiantes-con-discapacidad).