| Code |
15648
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| Year |
2
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| Semester |
S1
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| ECTS Credits |
6
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| Workload |
TP(45H)
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| Scientific area |
Mathematics
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Entry requirements |
Does not have.
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Learning outcomes |
In this curricular unit are introduced the basic concepts of the theory of Banach and Hilbert spaces.
At the end of the curricular unit the students should be able to:
- apply the basic theory of complete metric spaces;
- identify the main normed spaces;
- apply the theory of the normed spaces and of the Banach spaces;
- identify the main spaces with inner product;
- apply the theory of spaces with an inner product and of Hilbert spaces;
- know and use the main theorems of functional analysis: Hahn-Banach, Banach-Steinhaus, open application and closed graph;
- apply the definition of spectrum of an operator;
- know and use the main properties of the spectrum of an operator.
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Syllabus |
1. Revision of complete metric spaces
2. Normed spaces and Banach spaces
2.1 Definition, elementary properties and examples
2.2 Continuous linear operators
2.3 Functionals and dual space
2.4 Finite-dimensional Banach spaces
2.5 Compactness and Riesz Lemma
3. Inner product spaces and Hilbert spaces
3.1 Definition, elementary properties and examples
3.2 Orthogonal complement and ortogonal projections
3.3 Orthonormal sets
3.4 Functionals in Hilbert spaces
3.5 Adjoint operator
4. Fundamental theorems of Functional Analysis
4.1 Zorn’s Lemma
4.2 Hahn-Banach theorem
4.3 Dual operator
4.4 Reflexive spaces
4.5 Banach-Steinhaus theorem
4.6 Open mapping theorem and closed graph theorem
5. Spectral theory
5.1 Resolvent and spectrum of an operator
5.2 Spectral properties
5.3 Spectrum of compact operators
5.4 Spectrum of self-adjoint operators
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Main Bibliography |
Conway, J. B. (2013). A course in functional analysis. Springer Science & Business Media.
Giles, J. R. (2000). Introduction to the analysis of normed linear spaces. Cambridge University Press.
Kreyszig, E. (1978). Introductory functional analysis with applications New York: wiley.
Michel, A. N., & Herget, C. J. (2009). Algebra and analysis for engineers and scientists. Springer Science & BusinessMedia.
Rynne, B., & Youngson, M. A. (2011). Análise Funcional Linear. Coleção Ensino da Ciência e Tecnologia. IST Press.
Taylor, A. E., & Lay, D. C. (1986). Introduction to functional analysis. Krieger Publishing
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Teaching Methodologies and Assessment Criteria |
The curricular unit is organized in theoretical-practical lessons. At the beginning of each lesson the teacher makes a theoretical presentation of the subject and then the students solve exercises from worksheets.
The continuous evaluation consists of two written tests of 10 points each. If T1 and T2 are the grades of the first and second tests, respectively, the final grade is calculated as follows:
- if T1+T2 is less than 17.5, the final grade will be the rounding of T1+T2;
- if T1+T2 is greater than or equal to 17.5, the student must do an oral exam; to the oral exam is given a grade PO between 0 and 20 points; the final grade will be the rounding of max{17,(T1+T2+PO)/2}.
The students with a final grade greater than or equal to 10 points will pass the course.
Only the students with 50% of attendance will be admitted to the final exam.
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Language |
Portuguese. Tutorial support is available in English.
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